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 needs to increase 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 triple—to increase energy 9-fold requires tripling the amplitude because 3² = 9. Choice A is correct because it accurately uses the square root relationship to find that increasing energy 9-fold requires amplitude to triple (√9 = 3). Choice B incorrectly reverses the relationship, claiming amplitude must increase by a factor of 9 when energy increases 9-fold, when actually it's the reverse: amplitude must increase by √9 = 3 to achieve a 9-fold energy increase—this confusion between squaring and square root is a common error. 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). Real-world applications of E ∝ A²: to make a rope wave carry 9 times more energy, you only need to shake it 3 times as hard (triple the amplitude)—this is why it's physically possible to dramatically increase wave energy with manageable increases in effort, as the energy scales faster than the physical motion required to create it.

This question tests understanding of the relationship between wave amplitude and sound intensity. 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 2A₁, we calculate the intensity change using the squared relationship: I_new/I_original = (A_new/A_original)² = (2A₁/A₁)² = 2² = 4, so the intensity 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 $10^1$.5 in total energy released).

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 amplitude doubling: When amplitude doubles from A to 2A, we calculate the energy change using the squared relationship: E_new/E_original = (A_new/A_original)² = (2A/A)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 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 $10^1$.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.

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: Wave A with amplitude A_A = 0.20 mm and Wave B with amplitude A_B = 0.10 mm have intensity ratio I_A/I_B = (A_A/A_B)² = (0.20/0.10)² = 2² = 4, meaning Wave A carries 4 times as much intensity as Wave B despite having only twice the amplitude. Choice B is correct because it properly applies the squared relationship I ∝ A² to show that when Wave A has twice the amplitude of Wave B, it has 4 times the intensity. Choice A incorrectly treats the relationship as linear (I ∝ A), claiming that doubling amplitude doubles intensity, when actually the relationship is quadratic (I ∝ A²) so doubling amplitude quadruples intensity—this is why the answer is a factor of 4, not 2. 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 doubles, I increases by 2² = 4), (2) if asked how amplitude changes when intensity changes, take the square root of the intensity factor (if I increases 4-fold, A increases by √4 = 2), 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 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; 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.

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 $10^1$.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.

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 1.5 m to 3.0 m, we calculate the energy change using the squared relationship: E₂/E₁ = (A₂/A₁)² = (3.0/1.5)² = 2² = 4, so the energy increases by a factor of 4 (quadruples), not by a factor of 2—this is why a water wave that is twice as high carries 4 times as much energy. 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). 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 $10^1$.5 in total energy released).

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 $10^1$.5 in total energy released).

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 decreases from A₁ to 0.2 A₁, we calculate the energy change using the squared relationship: E₂/E₁ = (A₂/A₁)² = (0.2 A₁/A₁)² = (0.2)² = 0.04, so the energy decreases to a factor of 0.04 (1/25th), not by a factor of 0.2—this is why reducing amplitude to one-fifth in a seismic wave drastically reduces its energy. 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 0.2 amplitude means 0.2 energy, when actually the relationship is quadratic (E ∝ A²) so 0.2 amplitude means 0.04 energy—this is why the answer is a factor of 0.04, not 0.2. 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 $10^1$.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.

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 $10^1$.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.

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 $10^1$.5 in energy scaling).