Two water waves in the same region of the ocean have the same wavelength and speed. Wave A has amplitude (wave height above equilibrium) AA=0.50m and Wave B has amplitude AB=1.50m. If wave energy is proportional to amplitude squared (E∝A2), what is the ratio EB/EA?
Answer: 9
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. Wave A with amplitude AA = 0.50 m and Wave B with amplitude AB = 1.50 m have energy ratio EB/EA = (AB/AA)² = (1.50/0.50)² = 3² = 9, meaning Wave B carries 9 times as much energy as Wave A despite having only 3 times the amplitude. Choice B is correct because it properly applies the squared relationship E ∝ A² to show energy factor is square of amplitude factor. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that tripling amplitude triples energy, when actually the relationship is quadratic (E ∝ A²) so tripling amplitude increases energy 9-fold—this is why the answer is a factor of 9, not 3. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why increasing speaker volume slightly requires significantly more power, why tsunamis with modest height increase carry devastating energy, and why a magnitude 7 earthquake (10× amplitude of magnitude 6) releases roughly 32× more energy (actually 10² in displacement but ~32× in total energy released).
A loudspeaker produces a sound wave at a fixed frequency. The sound energy transferred to the air is proportional to the square of the wave's amplitude: E=kA2. If the speaker's amplitude increases from A1 to A2=2A1, by what factor does the energy change?
Answer: Increases by a factor of 4
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. When amplitude doubles from A₁ to A₂ = 2A₁, we calculate the energy change using the squared relationship: E₂/E₁ = (A₂/A₁)² = (2A₁/A₁)² = 2² = 4, so the energy increases by a factor of 4 (quadruples), not by a factor of 2—this is why a sound that is twice as loud in amplitude requires 4 times as much energy to produce. Choice B is correct because it properly applies the squared relationship E ∝ A² to show energy factor is square of amplitude factor. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that doubling amplitude doubles energy, when actually the relationship is quadratic (E ∝ A²) so doubling amplitude quadruples energy—this is why the answer is a factor of 4, not 2. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why increasing speaker volume slightly requires significantly more power, why tsunamis with modest height increase carry devastating energy, and why a magnitude 7 earthquake (10× amplitude of magnitude 6) releases roughly 32× more energy (actually 10² in displacement but ~32× in total energy released).
Two ocean surface waves pass a buoy. Wave A has amplitude AA=1.5m and Wave B has amplitude AB=0.5m (same wavelength and speed). If energy is proportional to amplitude squared (E∝A2), what is the ratio EA/EB?
Answer: 9
This question tests understanding of the relationship between wave amplitude and wave energy in ocean waves. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. For comparing waves: Wave A with amplitude AA = 1.5 m and Wave B with amplitude AB = 0.5 m have energy ratio EA/EB = (AA/AB)² = (1.5/0.5)² = 3² = 9, meaning Wave A carries 9 times as much energy as Wave B despite having only three times the amplitude. Choice B is correct because it accurately calculates that tripling the amplitude increases energy 9-fold, properly applying the squared relationship E ∝ A². Choice A incorrectly treats the relationship as linear (E ∝ A), claiming the ratio is 3, when actually the relationship is quadratic (E ∝ A²) so the ratio is 9—this distinction is crucial because it determines that tripling amplitude increases energy 9-fold not 3-fold. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: if asked how energy changes when amplitude changes, square the amplitude factor (if A triples, E increases by 3² = 9), and always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear. Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why larger ocean waves can cause much more erosion and damage to coastlines.
A loudspeaker produces a 500 Hz sound wave in air. With the frequency held constant, the wave energy transferred is proportional to the square of the amplitude: E∝A2. If the speaker's sound-wave amplitude increases from A1=0.20mm to A2=0.60mm, by what factor does the wave energy change?
Answer: Increases by a factor of 9
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. When amplitude increases from 0.20 mm to 0.60 mm, we calculate the energy change using the squared relationship: E₂/E₁ = (A₂/A₁)² = (0.60/0.20)² = 3² = 9, so the energy increases by a factor of 9, which is why increasing the speaker's vibration extent by three times requires nine times more energy to produce the sound wave. Choice B is correct because it properly applies the squared relationship E ∝ A² to show the energy factor is the square of the amplitude factor. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that tripling amplitude increases energy by a factor of 3, when actually the relationship is quadratic (E ∝ A²) so tripling amplitude increases energy nine-fold—this is why the answer is a factor of 9, not 3. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A triples, E increases by 3² = 9), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why increasing speaker volume slightly requires significantly more power, why tsunamis with modest height increase carry devastating energy, and why a magnitude 7 earthquake (10× amplitude of magnitude 6) releases roughly 32× more energy (actually 101.5 in energy scaling).
A seismic sensor records two surface waves from different earthquakes at the same station. Event 2 produces a ground-displacement amplitude that is 5 times larger than Event 1 (A2=5A1), with similar frequency content. Using E∝A2, what is the energy ratio E2/E1 carried by the wave?
Answer: 25
This question tests understanding of the relationship between wave amplitude and wave energy in seismic waves. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. For comparing waves: Event 1 with amplitude A₁ and Event 2 with amplitude A₂ = 5A₁ have energy ratio E₂/E₁ = (A₂/A₁)² = (5)² = 25, meaning Event 2 carries 25 times as much energy as Event 1 despite having only five times the amplitude. Choice C is correct because it accurately calculates that increasing amplitude by a factor of 5 increases energy 25-fold, properly applying the squared relationship E ∝ A². Choice B incorrectly treats the relationship as linear (E ∝ A), claiming the ratio is 5 (or perhaps 10, miscalculating), when actually the relationship is quadratic (E ∝ A²) so the ratio is 25—this distinction is crucial because it determines that a 5-fold amplitude increase means 25-fold energy increase. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: if asked how energy changes when amplitude changes, square the amplitude factor (if A increases by 5, E increases by 5² = 25), and always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear. Real-world applications of E ∝ A²: in earthquakes, each step up in Richter scale represents 10× more amplitude but about 32× more energy (roughly 101.5); in water waves, doubling wave height quadruples the energy, explaining why large tsunami waves are so devastating despite not looking enormously tall when viewed from space.
A lab measures the energy carried by waves on a string and finds that Wave 2 carries 16 times as much energy as Wave 1. Assuming E∝A2 for these string waves, what is the amplitude ratio A2/A1?
Answer: 4
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. For energy increasing: If energy increases by a factor of 16, we solve for amplitude change: E₂/E₁ = (A₂/A₁)² = 16, taking square root of both sides gives A₂/A₁ = √16 = 4, so amplitude must increase by a factor of 4—this shows that to increase energy 16-fold requires only quadrupling the amplitude because 4² = 16. Choice B is correct because it accurately uses the square root relationship to find that when energy increases by a factor of 16, amplitude increases by a factor of √16 = 4. Choice C makes an arithmetic error in taking the square root, incorrectly calculating that √16 = 8 instead of 4, or perhaps confuses the problem by thinking if E₂ = 16E₁ then A₂ = 8A₁, when the correct relationship requires taking the square root: A₂/A₁ = √(E₂/E₁) = √16 = 4. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 16-fold, A increases by √16 = 4), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why increasing speaker volume slightly requires significantly more power, why tsunamis with modest height increase carry devastating energy, and why a magnitude 7 earthquake (10× amplitude of magnitude 6) releases roughly 32× more energy.
A sound wave at fixed frequency travels through air. The wave's amplitude is increased by a factor of 3 while all other conditions remain the same. Using E∝A2, by what factor does the energy transported (or intensity) change?
Answer: It increases by a factor of 9.
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. When amplitude increases from A to 3A, we calculate the energy change using the squared relationship: E_new/E_original = (A_new/A_original)² = (3A/A)² = 3² = 9, so the energy increases by a factor of 9 (9 times), not by a factor of 3—this is why a sound that is three times as loud in amplitude requires 9 times as much energy to produce. Choice C is correct because it properly applies the squared relationship E ∝ A² to show energy factor is square of amplitude factor. Choice B incorrectly treats the relationship as linear (E ∝ A), claiming that tripling amplitude triples energy, when actually the relationship is quadratic (E ∝ A²) so tripling amplitude increases energy 9-fold—this is why the answer is a factor of 9, not 3. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Real-world applications of E ∝ A²: in sound, doubling the amplitude creates a sound that's 4 times more intense but only perceived as slightly louder due to logarithmic human hearing; in earthquakes, each step up in Richter scale represents 10× more amplitude but about 32× more energy (roughly 101.5); in water waves, doubling wave height quadruples the energy, explaining why large tsunami waves are so devastating despite not looking enormously tall when viewed from space.
A flashlight and a spotlight produce electromagnetic waves of the same frequency. At a point on a wall, the spotlight's electric-field amplitude is 5 times the flashlight's: As=5Af. If light intensity is proportional to amplitude squared (I∝A2), what is the intensity ratio Is/If at that point?
Answer: 25
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. For comparing waves: The spotlight with amplitude As = 5A_f (5 times the flashlight's amplitude) and the flashlight with amplitude Af have intensity ratio Is/If = (As/Af)² = (5A_f/Af)² = 5² = 25, meaning the spotlight delivers 25 times as much intensity despite having only 5 times the electric field amplitude. Choice C is correct because it properly applies the squared relationship I ∝ A² to show that when the spotlight has 5 times the amplitude of the flashlight, it has 25 times the intensity. Choice A incorrectly treats the relationship as linear (I ∝ A), claiming that having 5 times the amplitude means 5 times the intensity, when actually the relationship is quadratic (I ∝ A²) so 5 times the amplitude means 25 times the intensity—this is why the answer is 25, not 5. To solve amplitude-energy problems, remember the formula I₂/I₁ = (A₂/A₁)²: (1) if asked how intensity changes when amplitude changes, square the amplitude factor (if A increases 5-fold, I increases by 5² = 25), (2) if asked how amplitude changes when intensity changes, take the square root of the intensity factor (if I increases 25-fold, A increases by √25 = 5), and (3) always check your answer makes sense—larger amplitude must mean more intensity, and the relationship is stronger than linear (intensity grows faster than amplitude). Real-world applications of I ∝ A²: in light, doubling the electric field amplitude creates light that's 4 times more intense; this squared relationship explains why laser beams with slightly larger electric field amplitudes can deliver dramatically more power for cutting or welding, and why even small increases in electromagnetic wave amplitude can significantly increase the energy delivered for applications like wireless power transmission.
A sound wave in air has energy proportional to amplitude squared: E∝A2. A technician wants to double the energy output (E2=2E1) without changing frequency. By what factor must the amplitude change (A2/A1)?
Answer: 2
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. If energy increases by a factor of 2, we solve for amplitude change: E₂/E₁ = (A₂/A₁)² = 2, taking square root of both sides gives A₂/A₁ = √2 ≈ 1.414, so amplitude must increase by a factor of √2—to double energy requires only increasing amplitude by about 41% because (√2)² = 2. Choice B is correct because it accurately uses square root relationship to find amplitude change from energy change. Choice A incorrectly reverses the relationship, confusing the energy factor (2) with the amplitude factor (√2)—when energy doubles, amplitude increases only by √2 (not 2) because we must take the square root: A₂/A₁ = √(E₂/E₁) = √2 ≈ 1.414. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Real-world applications of E ∝ A²: in sound, doubling the amplitude creates a sound that's 4 times more intense but only perceived as slightly louder due to logarithmic human hearing; in earthquakes, each step up in Richter scale represents 10× more amplitude but about 32× more energy (roughly 101.5); in water waves, doubling wave height quadruples the energy, explaining why large tsunami waves are so devastating despite not looking enormously tall when viewed from space.
An electromagnetic wave (light) has intensity proportional to the square of the electric field amplitude: I∝A2. If the amplitude is reduced to half its original value (frequency unchanged), what happens to the intensity?
Answer: It becomes 1/4 as large
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you halve the amplitude (A → A/2), the energy decreases by a factor of 4 (E → E/4)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. When amplitude is reduced to half its original value (A → A/2), we calculate the intensity change using the squared relationship: I₂/I₁ = (A₂/A₁)² = (A/2 / A)² = (1/2)² = 1/4, so the intensity becomes 1/4 as large—halving the amplitude reduces intensity to one-quarter of its original value. Choice B is correct because it properly applies the squared relationship I ∝ A² to show that halving amplitude reduces intensity to 1/4. Choice A incorrectly treats the relationship as linear (I ∝ A), claiming that halving amplitude halves intensity, when actually the relationship is quadratic (I ∝ A²) so halving amplitude reduces intensity to one-quarter—this is why the answer is 1/4, not 1/2. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A is halved, E becomes (1/2)² = 1/4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E becomes 1/4, A becomes √(1/4) = 1/2), and (3) always check your answer makes sense—smaller amplitude must mean less energy, and the relationship is stronger than linear (energy decreases faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far the electric field oscillates (amplitude) and how much energy is stored in the field—both effects scale with amplitude, so total intensity scales as amplitude squared; this is why dimming a light by reducing the electric field amplitude to half produces a light that appears much dimmer than half brightness, as it actually has only 1/4 the intensity.
A sound wave's energy carried per unit time (and intensity at a fixed point) follows E∝A2. A speaker's amplitude control is increased so the amplitude becomes 1.5 times the original value, with frequency unchanged. By what factor does the energy carried by the wave change?
Answer: Increases by a factor of 2.25
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you increase amplitude by 1.5 times (A → 1.5A), the energy increases by a factor of 2.25 (E → 2.25E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. When amplitude becomes 1.5 times the original value, we calculate the energy change using the squared relationship: E₂/E₁ = (A₂/A₁)² = (1.5A/A)² = 1.5² = 2.25, so the energy increases by a factor of 2.25—increasing amplitude by 50% (factor of 1.5) increases energy by 125% (factor of 2.25). Choice B is correct because it properly applies the squared relationship E ∝ A² to show that increasing amplitude by a factor of 1.5 increases energy by 1.5² = 2.25. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that increasing amplitude by 1.5 times increases energy by 1.5 times, when actually the relationship is quadratic (E ∝ A²) so 1.5 times the amplitude means 2.25 times the energy—this is why the answer is a factor of 2.25, not 1.5. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A increases by 1.5, E increases by 1.5² = 2.25), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 2.25-fold, A increases by √2.25 = 1.5), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Real-world applications of E ∝ A²: in sound systems, increasing the speaker cone displacement by just 50% (1.5×) requires 2.25 times as much power—this explains why small increases in volume control can lead to significant increases in power consumption, and why audio amplifiers need substantial power reserves to handle even modest increases in output level.
A student sends waves on a rope with the same frequency and tension each time. The wave energy is proportional to amplitude squared: E∝A2. Trial 2 has 9 times the energy of Trial 1 (E2=9E1). By what factor did the amplitude change (A2/A1)?
Answer: 3
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. If energy increases by a factor of 9, we solve for amplitude change: E₂/E₁ = (A₂/A₁)² = 9, taking square root of both sides gives A₂/A₁ = √9 = 3, so amplitude must increase by a factor of 3—for example, to increase energy 9-fold requires tripling the amplitude because 3² = 9. Choice A is correct because it accurately uses square root relationship to find amplitude change from energy change. Choice B incorrectly reverses the relationship, confusing the energy factor (9) with the amplitude factor (3)—when energy increases 9-fold, amplitude increases only 3-fold (not 9-fold) because we must take the square root: A₂/A₁ = √(E₂/E₁) = √9 = 3. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Real-world applications of E ∝ A²: in sound, doubling the amplitude creates a sound that's 4 times more intense but only perceived as slightly louder due to logarithmic human hearing; in earthquakes, each step up in Richter scale represents 10× more amplitude but about 32× more energy (roughly 101.5); in water waves, doubling wave height quadruples the energy, explaining why large tsunami waves are so devastating despite not looking enormously tall when viewed from space.
A water wave in a wave tank has amplitude A1=0.20 m (measured from equilibrium to crest). A second wave in the same tank has energy per unit area that is 9 times larger than the first. Assuming wave energy is proportional to amplitude squared, E∝A2, what is the amplitude factor A2/A1?
Answer: 3
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. If energy increases by a factor of 9, we solve for amplitude change: E₂/E₁ = (A₂/A₁)² = 9, taking square root of both sides gives A₂/A₁ = √9 = 3, so amplitude must increase by a factor of 3—for example, to increase energy 4-fold requires only doubling the amplitude because 2² = 4. Choice B is correct because it accurately uses square root relationship to find amplitude change from energy change. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that 9-fold energy requires 9-fold amplitude, when actually the relationship is quadratic (E ∝ A²) so 9-fold energy requires only 3-fold amplitude—this is why the answer is a factor of 3, not 9. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why increasing speaker volume slightly requires significantly more power, why tsunamis with modest height increase carry devastating energy, and why a magnitude 7 earthquake (10× amplitude of magnitude 6) releases roughly 32× more energy (actually 101.5 in total energy released).
A light (electromagnetic) wave has intensity I∝A2. If the intensity increases by a factor of 16 (for example, a spotlight appears much brighter), by what factor did the wave's amplitude increase?
Answer: 4
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. If intensity increases by a factor of 16, we solve for amplitude change: I₂/I₁ = (A₂/A₁)² = 16, taking square root of both sides gives A₂/A₁ = √16 = 4, so amplitude must increase by a factor of 4—to increase intensity 16-fold requires only quadrupling the amplitude because 4² = 16. Choice B is correct because it accurately uses square root relationship to find amplitude change from intensity change. Choice D incorrectly reverses the relationship, confusing the intensity factor (16) with the amplitude factor (4)—when intensity increases 16-fold, amplitude increases only 4-fold (not 16-fold) because we must take the square root: A₂/A₁ = √(I₂/I₁) = √16 = 4. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why increasing speaker volume slightly requires significantly more power, why tsunamis with modest height increase carry devastating energy, and why a magnitude 7 earthquake (10× amplitude of magnitude 6) releases roughly 32× more energy (actually 10² in displacement but ~32× in total energy released).
Two water waves in a wave tank have the same wavelength and speed. Wave 1 has height (amplitude) A1=0.20m and Wave 2 has A2=0.60m. If wave energy is proportional to amplitude squared (E∝A2), how do their energies compare (ratio E2/E1)?
Answer: 9
This question tests understanding of the relationship between wave amplitude and wave energy in water waves. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. For comparing waves: Wave 1 with amplitude A₁ = 0.20 m and Wave 2 with amplitude A₂ = 0.60 m have energy ratio E₂/E₁ = (A₂/A₁)² = (0.60/0.20)² = 3² = 9, meaning Wave 2 carries 9 times as much energy as Wave 1 despite having only three times the amplitude. Choice B is correct because it accurately calculates that tripling the amplitude increases energy 9-fold, properly applying the squared relationship E ∝ A². Choice A incorrectly treats the relationship as linear (E ∝ A), claiming the ratio is 3, when actually the relationship is quadratic (E ∝ A²) so the ratio is 9—this distinction is crucial because it determines that tripling amplitude increases energy 9-fold not 3-fold. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: if asked how energy changes when amplitude changes, square the amplitude factor (if A triples, E increases by 3² = 9), and always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear. Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why tsunamis with modest height increase carry devastating energy.
A sound wave at fixed frequency has intensity proportional to amplitude squared: I∝A2. A microphone measures intensity I1 when the amplitude is A1. If the measured intensity becomes I2=9I1, by what factor did the amplitude change?
Answer: Amplitude increased by a factor of 3
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. For energy increasing: If intensity (energy-related) increases by a factor of 9, we solve for amplitude change: I₂/I₁ = (A₂/A₁)² = 9, taking square root of both sides gives A₂/A₁ = √9 = 3, so amplitude must increase by a factor of 3—for example, to increase energy 4-fold requires only doubling the amplitude because 2² = 4. Choice C is correct because it accurately uses square root relationship to find amplitude change from energy change. Choice A incorrectly assumes energy is proportional to amplitude squared but then fails to take the square root, claiming a factor of 9 for amplitude when it's actually √9 = 3. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why increasing speaker volume slightly requires significantly more power, why tsunamis with modest height increase carry devastating energy, and why a magnitude 7 earthquake (10× amplitude of magnitude 6) releases roughly 32× more energy (actually 101.5 in energy scaling).
A water wave near shore has energy proportional to amplitude squared: E=kA2. During a storm, the wave amplitude increases from A1=0.8m to A2=1.6m (frequency unchanged). By what factor does the wave energy change?
Answer: Increases by a factor of 4
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. When amplitude increases from A₁ = 0.8 m to A₂ = 1.6 m, we calculate the energy change using the squared relationship: E₂/E₁ = (A₂/A₁)² = (1.6/0.8)² = 2² = 4, so the energy increases by a factor of 4 (quadruples)—this is why storm waves with double the height carry four times the destructive power. Choice C is correct because it properly applies the squared relationship E ∝ A² to show energy factor is square of amplitude factor. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that doubling amplitude doubles energy, when actually the relationship is quadratic (E ∝ A²) so doubling amplitude quadruples energy—this is why the answer is a factor of 4, not 2. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Real-world applications of E ∝ A²: in sound, doubling the amplitude creates a sound that's 4 times more intense but only perceived as slightly louder due to logarithmic human hearing; in earthquakes, each step up in Richter scale represents 10× more amplitude but about 32× more energy (roughly 101.5); in water waves, doubling wave height quadruples the energy, explaining why large tsunami waves are so devastating despite not looking enormously tall when viewed from space.
A sound wave at a fixed frequency is recorded by a microphone. Recording 1 has amplitude A1, and Recording 2 has amplitude A2=0.50A1. If E∝A2, what is the ratio E2/E1?
Answer: E2/E1=0.25
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you halve the amplitude (A → 0.5A), the energy decreases by a factor of 4 (E → 0.25E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. When amplitude decreases from A₁ to A₂ = 0.50A₁, we calculate the energy change using the squared relationship: E₂/E₁ = (A₂/A₁)² = (0.50A₁/A₁)² = (0.50)² = 0.25, so the energy decreases to one-fourth of its original value—this means Recording 2 with half the amplitude has only 25% of the energy of Recording 1. Choice B is correct because it properly applies the squared relationship E ∝ A² to show that halving amplitude reduces energy to 0.25 (one-fourth). Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that halving amplitude halves energy to 0.50, when actually the relationship is quadratic (E ∝ A²) so halving amplitude reduces energy to 0.25—this is why the answer is 0.25, not 0.50. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Real-world applications of E ∝ A²: in sound, doubling the amplitude creates a sound that's 4 times more intense but only perceived as slightly louder due to logarithmic human hearing; in earthquakes, each step up in Richter scale represents 10× more amplitude but about 32× more energy (roughly 101.5); in water waves, doubling wave height quadruples the energy, explaining why large tsunami waves are so devastating despite not looking enormously tall when viewed from space.
A student compares two waves on the same rope and claims: "If I double the amplitude, I double the energy." Which statement best describes the correct amplitude–energy relationship for these rope waves (holding other conditions constant)?
Answer: Energy is proportional to the square of amplitude: E∝A2.
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. The mathematical relationship is E = kA² where k is a constant depending on the medium, frequency, and other wave properties—the key insight is the squared dependence, which appears in intensity of sound (I ∝ A²), power in water waves (P ∝ A²), and brightness of light (brightness ∝ A²), all reflecting the fundamental E ∝ A² relationship. Choice B is correct because it correctly identifies that energy is proportional to amplitude squared. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that doubling amplitude doubles energy, when actually the relationship is quadratic (E ∝ A²) so doubling amplitude quadruples energy—this is why the student's claim is wrong. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Real-world applications of E ∝ A²: in sound, doubling the amplitude creates a sound that's 4 times more intense but only perceived as slightly louder due to logarithmic human hearing; in earthquakes, each step up in Richter scale represents 10× more amplitude but about 32× more energy (roughly 101.5); in water waves, doubling wave height quadruples the energy, explaining why large tsunami waves are so devastating despite not looking enormously tall when viewed from space.
A lab handout states that for waves of the same type traveling in the same medium (with frequency unchanged), the energy carried by the wave depends on amplitude according to E=kA2 for some constant k. Which statement correctly describes the relationship between amplitude and energy?
Answer: Energy is proportional to amplitude squared: E∝A2
This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. For identifying the relationship: The mathematical relationship is E = kA² where k is a constant depending on the medium, frequency, and other wave properties—the key insight is the squared dependence, which appears in intensity of sound (I ∝ A²), power in water waves (P ∝ A²), and brightness of light (brightness ∝ A²), all reflecting the fundamental E ∝ A² relationship stated in the problem. Choice B is correct because it correctly identifies that energy is proportional to amplitude squared, matching the given formula E = kA². Choice A incorrectly states a linear relationship E ∝ A, which would mean doubling amplitude doubles energy, when the correct relationship E ∝ A² means doubling amplitude quadruples energy—this distinction is crucial because it determines that increasing amplitude has a much stronger effect on energy than a simple proportional relationship would suggest. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 4-fold, A increases by √4 = 2), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this universal E ∝ A² relationship applies to all wave types because it emerges from the fundamental physics of oscillating systems storing energy in both kinetic and potential forms.
