Wave Interference and Standing Waves - AP Physics 2

What is an antinode?

A point where the amplitude of a standing wave is maximum. Point of constructive interference in standing wave patterns.
Identify the condition for destructive interference.

Path difference is an odd multiple of $\frac{\text{\textit{λ}}}{2}$ . Waves arrive out of phase when path difference is half-wavelengths.
What determines the frequency of a standing wave in a string?

String length, tension, and mass per unit length. These parameters determine the fundamental frequency and harmonics.
State the formula for the fundamental frequency of a string.

$f_1 = \frac{v}{2L}$ , where $v$ is wave speed, $L$ is length. The lowest frequency mode for a vibrating string.
What is a standing wave?

A wave that appears to be stationary, with nodes and antinodes. Formed by two waves traveling in opposite directions interfering.
Calculate the length of a string for a fundamental frequency of $440 \text{ Hz}$ , $v = 330 \text{ m/s}$ .

Length $L = 0.375 \text{ m}$ . Using $L = \frac{v}{2f_1} = \frac{330}{2 \times 440} = 0.375$ m.
What is the relationship between frequency and wavelength?

Inversely proportional: $f \text{\textit{λ}} = v$ .. Higher frequency means shorter wavelength for constant wave speed.
What is the phase difference for complete destructive interference?

Phase difference is $\text{\textit{π}}$ radians. Waves oscillate in perfect opposition.
Identify the condition for constructive interference.

Path difference is a multiple of the wavelength, $n \lambda$ . Waves arrive in phase when path difference equals whole wavelengths.
Calculate the third harmonic for $L = 1 \text{ m}$ , $v = 340 \text{ m/s}$ .

Third harmonic $f_3 = 510 \text{ Hz}$ . Using $f_3 = 3f_1 = 3 \cdot \frac{340}{2} = 510$ Hz.
Determine the beat frequency for $f_1 = 256 \text{ Hz}$ , $f_2 = 260 \text{ Hz}$ .

Beat frequency is $4 \text{ Hz}$ . Using $f_{beat} = |256 - 260| = 4$ Hz.
Calculate the frequency of the second harmonic for $L = 2 \text{ m}$ , $v = 340 \text{ m/s}$ .

$f_2 = 170 \text{ Hz}$ .. Second harmonic: $f_2 = 2f_1 = 2 \cdot \frac{340}{4} = 170$ Hz.
State the relationship between nodes and antinodes.

Nodes are points of no displacement; antinodes have maximum displacement. Nodes and antinodes are separated by $\frac{\lambda}{4}$ .
State the relationship between wave speed and medium.

Wave speed depends on the medium's properties. Wave speed is independent of frequency for a given medium.
Define constructive interference.

Occurs when wave amplitudes add to create a larger amplitude. Waves are in phase, causing amplification of the resultant wave.
If a wave has an amplitude $A$ , what is its intensity proportional to?

Intensity is proportional to $A^2$ . Wave energy is proportional to amplitude squared.
What is the impact of damping on a standing wave?

Damping reduces wave amplitude over time. Energy dissipation causes gradual decay of oscillation amplitude.
Calculate the fundamental frequency for $L = 0.5 \text{ m}$ , $v = 300 \text{ m/s}$ .

$f_1 = 300 \text{ Hz}$ . Using $f_1 = \frac{v}{2L} = \frac{300}{2 \times 0.5} = 300$ Hz.
Find the wavelength of a wave given $v = 340 \text{ m/s}$ , $f = 170 \text{ Hz}$ .

Wavelength $\text{\textit{λ}} = 2 \text{ m}$ . Using $\lambda = \frac{v}{f} = \frac{340}{170} = 2$ m.
What is formed when two waves of the same frequency and amplitude meet in phase?

A resultant wave with doubled amplitude. Perfect constructive interference produces maximum possible amplitude.
What is the effect of tension on wave speed in a string?

Increased tension increases wave speed. Higher tension provides greater restoring force, increasing wave speed.
What is the formula for the nth harmonic frequency?

$f_n = n\frac{v}{2L}$ , $n = 1, 2, 3, \text{...}$ . Each harmonic is an integer multiple of the fundamental frequency.
What is the phase difference for complete constructive interference?

Phase difference is $0$ or $2\text{\textit{π}}$ radians. Waves oscillate in perfect synchronization.
Identify the effect of boundary conditions on standing waves.

Nodes at fixed boundaries; antinodes at open boundaries. Boundary conditions determine the standing wave pattern.
If two waves meet out of phase, what is the result?

Destructive interference. Out of phase waves interfere destructively, reducing amplitude.
What is a node?

A point along a standing wave with zero amplitude. Point of destructive interference in standing wave patterns.
What is the equation for wave speed?

$v = f\text{\textit{λ}}$ , where $f$ is frequency, $\text{\textit{λ}}$ is wavelength. Fundamental relationship between wave properties.
Identify the type of interference with a phase difference of $\frac{\text{\textit{π}}}{2}$ .

Neither complete constructive nor destructive interference. Partial interference occurs with phase differences between 0 and $\pi$ .
Define destructive interference.

Occurs when wave amplitudes cancel to create a smaller amplitude. Waves are out of phase, reducing the resultant amplitude.
What is the speed of a wave with $\text{\textit{λ}} = 3 \text{ m}$ , $f = 100 \text{ Hz}$ ?

Wave speed $v = 300 \text{ m/s}$ .. Using $v = f\lambda = 100 \times 3 = 300$ m/s.
Calculate wave speed for $\text{\textit{λ}} = 5 \text{ m}$ , $f = 50 \text{ Hz}$ .

Wave speed $v = 250 \text{ m/s}$ . Using $v = f\lambda = 50 \times 5 = 250$ m/s.
Which physical quantity remains unchanged in standing waves?

The frequency of the wave remains unchanged. Standing waves maintain constant frequency as amplitude varies spatially.
How do you calculate beat frequency?

Beat frequency is $|f_1 - f_2|$ , where $f_1$ and $f_2$ are frequencies. Beat frequency equals the absolute difference between interfering frequencies.
What is the phase shift for waves in antiphase?

Phase difference of $\text{\textit{π}}$ radians. Antiphase means waves are exactly out of phase by half cycle.
Determine the first overtone frequency for $f_1 = 220 \text{ Hz}$ .

First overtone frequency is $440 \text{ Hz}$ . First overtone is the second harmonic: $f_2 = 2f_1$ .
What happens to wave speed if tension in a string is halved?

Wave speed decreases by factor of $\frac{1}{\sqrt{2}}$ . Wave speed is proportional to $\sqrt{T}$ , so halving tension reduces speed.
What is the principle of superposition?

The net displacement is the sum of individual displacements. This describes wave superposition where effects combine algebraically.
What is the formula for intensity of a wave?

$I = \frac{P}{A}$ , where $P$ is power, $A$ is the area. Intensity measures energy flow per unit area per unit time.

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