Wave Properties
A wave has a frequency of 50 Hz and travels at 340 m/s. What is its wavelength?
λ = v/f = 340 / 50 = 6.8 m
What are two features that distinguish a standing wave from a travelling wave?
A standing wave does not transfer energy along its length (a travelling wave does), and it has fixed points of zero displacement (nodes) and maximum displacement (antinodes) that stay in place, rather than a pattern that moves.
Define constructive and destructive interference in terms of path difference.
Constructive interference occurs where the path difference is a whole number of wavelengths (0, λ, 2λ, ...), so waves arrive in phase. Destructive interference occurs where the path difference is not a whole number of wavelengths, so waves arrive out of phase.
What is diffraction, and what property of waves does it demonstrate that particles do not?
Diffraction is the spreading of a wave as it passes through a gap or around an obstacle. It demonstrates that waves can bend around barriers and spread into regions particles could never reach in a straight line, a behaviour unique to waves.
A car sounds its horn as it drives towards a stationary observer. What happens to the observed frequency compared to the emitted frequency?
The observed frequency is higher than the emitted frequency.
Two waves have the same speed but different wavelengths. What can you say about their frequencies, and why?
Since v = fλ and v is fixed, f and λ are inversely proportional, the wave with the shorter wavelength must have the higher frequency.
A string fixed at both ends vibrates in its second harmonic. Describe the pattern of nodes and antinodes.
There are 3 nodes (one at each fixed end and one in the middle) and 2 antinodes, forming one full wavelength along the string's length.
Two speakers emit sound waves in phase. At a point where the path difference is exactly one wavelength, what type of interference occurs?
Constructive interference, because a path difference of one full wavelength means the waves arrive in phase.
Describe how the diffraction pattern changes as the gap width is made narrower relative to the wavelength.
As the gap becomes narrower relative to the wavelength, the wave spreads out more after passing through, approaching a full semicircular spreading pattern when the gap is similar in size to or smaller than the wavelength.
Explain, in terms of wavefronts, why an approaching source causes an increase in observed frequency.
As the source moves towards the observer, each successive wavefront is emitted from a position closer to the observer than the last, compressing the wavefronts in front of the source. This shortens the effective wavelength reaching the observer, and since v = fλ with v fixed, a shorter wavelength means a higher observed frequency.
A student says: "Increasing the amplitude of a wave increases its speed." Evaluate this statement using the wave equation and energy relationships.
False. Wave speed is set by the properties of the medium, not by amplitude or frequency. Amplitude affects the energy carried by the wave, not how fast it travels. v = fλ shows speed depends on f and λ, and increasing amplitude changes neither.
Explain, in terms of reflected waves, why a standing wave forms in a pipe closed at one end.
A wave sent down the pipe reflects off the closed end. The incident and reflected waves, travelling in opposite directions with the same frequency and wavelength, superpose. At certain points they are always in out of phase (destructive interference - a node) and at others always in phase (constructive interference - an antinode), producing a fixed interference pattern rather than a travelling one.
Explain why two coherent sources are needed to produce a stable interference pattern, and what would happen if the sources were not coherent.
Coherent sources have the same frequency and a constant phase relationship, so the phase difference at any given point in space stays constant over time, producing a fixed, observable pattern of constructive and destructive interference. Without coherence, the phase relationship between the sources is not fixed, so the interference pattern shifts too quickly to be observed.
Explain why low-frequency sound diffracts around corners more noticeably than high-frequency sound.
Low-frequency sound has a longer wavelength (since v = fλ and v is fixed). Noticeable diffraction occurs when the wavelength is comparable to or larger than the size of the gap or obstacle, so the long wavelengths of low-frequency sound diffract significantly around typical obstacles (like door frames or corners), while the much shorter wavelengths of high-frequency sound do not spread as much and travel more directionally.
An ambulance passes a stationary observer. Describe and explain the change in pitch heard as the ambulance approaches, passes, and moves away.
As it approaches, the pitch is higher than the true emitted pitch because the source moving towards the observer compresses the wavefronts, raising the observed frequency. At the instant it passes, the pitch matches the true emitted frequency (zero relative velocity). As it moves away, the pitch drops below the true frequency because the source moving away stretches the wavefronts, lowering the observed frequency.
Explain why the speed of a wave depends on the medium it travels through, not on its frequency or amplitude, using the example of sound travelling from air into water.
Wave speed depends on the physical properties of the medium, because these determine how quickly a disturbance can be passed from particle to particle. Water is much stiffer than air, so sound travels faster in water (~1480 m/s) than in air (~340 m/s), regardless of the frequency or amplitude of the source. Frequency is fixed by the source and stays the same crossing the boundary, so wavelength must change to keep v = fλ consistent.
A guitar string produces a fundamental frequency of 220 Hz. Explain what physically must happen to the string for it to produce its third harmonic instead, and calculate that frequency.
The string must vibrate in three equal segments instead of one, meaning three loops separated by extra nodes fit along its fixed length — this requires a faster oscillation, achieved by exciting the string in a way that supports that pattern (e.g. lightly touching a node point).
Third harmonic frequency = 3 × 220 = 660 Hz.
Two coherent speakers are placed 2.0 m apart, emitting sound at 850 Hz in phase (speed of sound = 340 m/s).
Calculate the wavelength, and determine the path difference at the point where third-order constructive interference occurs.
λ = v/f = 340/850 = 0.4 m.
Constructive interference occurs where path difference = nλ, so third order (n = 3) gives a path difference of 3 × 0.4 = 1.2 m.
Monochromatic light from a laser is shone through a diffraction grating with 300 lines per millimetre. The second-order maximum is observed at an angle of 20.5° from the central maximum. Calculate the wavelength of the light.
d = 1/300 mm = 3.33 × 10⁻⁶ m.
Using dsinθ = nλ: λ = d sinθ / n = (3.33 × 10⁻⁶ × sin20.5°) / 2 = (3.33 × 10⁻⁶ × 0.350) / 2 = 5.83 × 10⁻⁷ m ≈ 583 nm.
Explain why the Doppler effect occurs due to relative motion between source and observer, using the case of a stationary source and a moving observer, and explain why the wavelength does not change in this case (unlike a moving source).
With a stationary source, the wavefronts are emitted symmetrically and spaced at the true, unchanged wavelength. An observer moving towards the source, crosses more wavefronts per second than if they were stationary because they are closing the gap between themselves and each successive wavefront. The observed frequency increases even though the wavelength in the medium is unchanged. This shows the Doppler shift arises from relative motion changing how often wavefronts are encountered, not always from a physical change in wavelength.
A wave's frequency stays constant as it moves from a shallow to a deep section of water where its speed increases. Explain what happens to the wavelength, and justify how this can be true given that a wave carries energy without transporting mass.
Since f is fixed and v increases, λ must increase (v = fλ). This is possible because a wave transfers energy through a medium via oscillation of particles about a fixed position, not by moving matter along with it. Only the pattern of disturbance passing through changes shape, spacing the crests further apart as it speeds up.
Two identical strings under different tensions are plucked. Justify, using the wave speed on a string and standing wave formation, why the string under higher tension produces a higher-pitched fundamental note even though both strings have the same length.
(v = √(T/mass per unit length)
Wave speed on a string increases with tension. For a fixed length L, the fundamental standing wave requires λ = 2L regardless of tension, so f = v/λ = v/2L. Since higher tension increases v while λ stays fixed by the string's length, the fundamental frequency f increases, producing a higher pitch.
Two coherent speakers are 3.0 m apart. A student walks along a line 6.0 m from the midpoint between them and finds the first point of destructive interference where the path lengths from the two speakers are 6.11 m and 5.43 m.
Calculate the frequency of the sound (speed of sound = 340 m/s), given this is the first-order destructive interference point.
Path difference = 6.11 − 5.43 = 0.68 m.
First-order destructive interference occurs at a path difference of 0.5λ, so λ = 0.68 / 0.5 = 1.36 m.
f = v/λ = 340/1.36 = 250 Hz.
White light is shone through the same diffraction grating (300 lines per mm). Calculate the angular width of the first-order spectrum (i.e. the angle between the red end and the violet end), and explain why, unlike a single-wavelength source, white light produces a spread of angles at each order rather than a single sharp maximum.
d = 3.33 × 10⁻⁶ m. For violet (400 nm): sinθ = nλ/d = (1 × 400×10⁻⁹)/3.33×10⁻⁶ = 0.120, so θ = 6.90°.
For red (700 nm): sinθ = (1 × 700×10⁻⁹)/3.33×10⁻⁶ = 0.210, so θ = 12.1°. Angular width = 12.1° − 6.90° = 5.2°.
This spread happens because each wavelength satisfies dsinθ = nλ at a different angle for the same order n. Since white light contains a continuous range of wavelengths, each one diffracts to a slightly different angle, spreading the single maximum a monochromatic source would produce into a continuous spectrum.
A distant galaxy's light is observed to be redshifted. Using the Doppler effect principle, explain what this tells astronomers about the galaxy's motion relative to Earth, and why this same principle applies to both sound and light despite fundamental differences between the two types of waves.
Redshift means the observed wavelength of light is longer (frequency lower) than the wavelength emitted, which indicates the galaxy is moving away from Earth. The Doppler principle applies to any wave phenomenon because it depends only on the general relationship between relative motion of source and observer and the spacing of wavefronts (v = fλ), not on the specific medium or nature of the wave; this is why it applies to both sound (a mechanical wave needing a medium) and light (an electromagnetic wave that does not).