Sound Waves — Revision Notes

⚡ 30-Second Revision

Nature: — Mechanical, Longitudinal wave. Needs medium. Cannot travel in vacuum.
Speed: — $v = flambda$. In gas: $v = sqrt{gamma RT/M}$. $v propto sqrt{T}$.
Perception:

* Pitch $leftrightarrow$ Frequency ($f$) * Loudness $leftrightarrow$ Amplitude ($A$), Intensity ($I$). $\beta = 10 log_{10}(I/I_0)$. * Quality $leftrightarrow$ Waveform (overtones).

Phenomena: — Reflection (echo), Refraction, Diffraction, Interference.
Beats: — $f_{\text{beat}} = |f_1 - f_2|$.
Doppler Effect: — f' = f left( \frac{v \pm v_o}{v \mp v_s} \right). (+ $v_o$ towards, - $v_s$ towards).
Standing Waves:

* String (fixed ends): $f_n = n(v/2L)$, all harmonics. * Open Pipe: $f_n = n(v/2L)$, all harmonics. * Closed Pipe: $f_n = (2n-1)(v/4L)$, only odd harmonics.

2-Minute Revision

Sound waves are mechanical, longitudinal waves that propagate through a medium via compressions and rarefactions. They cannot travel in a vacuum. The speed of sound ($v$) is determined by the medium's elasticity and density, increasing with temperature in gases ($v propto sqrt{T}$).

The fundamental wave equation is $v = flambda$, where $f$ is frequency (pitch) and $lambda$ is wavelength. Amplitude relates to loudness (intensity $I propto A^2$), and quality depends on overtones. When sound moves between media, its frequency remains constant, but speed and wavelength change.

Key phenomena include reflection (echoes), refraction, diffraction, and interference. Beats occur when two slightly different frequencies interfere, with beat frequency $f_{\text{beat}} = |f_1 - f_2|$.

The Doppler effect describes the apparent change in frequency due to relative motion between source and observer, given by f' = f left( \frac{v pm v_o}{v mp v_s} \right). Standing waves are crucial for musical instruments: in strings and open pipes, all harmonics ($f_n = n(v/2L)$) are present, while in closed pipes, only odd harmonics ($f_n = (2n-1)(v/4L)$) are present.

Remember to use correct sign conventions for Doppler effect and distinguish between open and closed pipe harmonics.

5-Minute Revision

Sound waves are fundamental mechanical, longitudinal disturbances that require a material medium for propagation, unlike light. They consist of alternating regions of high pressure (compressions) and low pressure (rarefactions) that travel by particle oscillation parallel to the wave direction.

The speed of sound ($v$) is medium-dependent; in gases, it's given by $v = sqrt{gamma RT/M}$ and increases with the square root of absolute temperature ($v propto sqrt{T}$). Humidity also increases speed slightly.

The wave equation $v = flambda$ links speed, frequency ($f$, related to pitch), and wavelength ($lambda$). The amplitude of the wave determines its intensity ($I propto A^2$), which we perceive as loudness (measured in decibels, $\beta = 10 log_{10}(I/I_0)$).

The quality or timbre of a sound is due to the presence and relative strength of overtones.

Sound exhibits typical wave phenomena: reflection (causing echoes), refraction (bending due to speed changes), and diffraction (bending around obstacles). Interference occurs when waves superpose. A special case is beats, formed by the interference of two waves with slightly different frequencies ($f_1, f_2$), producing a pulsating sound with a beat frequency $f_{\text{beat}} = |f_1 - f_2|$.

The Doppler effect is crucial: it's the apparent change in frequency ($f'$) when there's relative motion between the source and observer. The general formula is f' = f left( \frac{v pm v_o}{v mp v_s} \right), where $v$ is sound speed, $v_o$ is observer speed, and $v_s$ is source speed. Remember the sign convention: 'towards' increases frequency (numerator +, denominator -), 'away' decreases frequency (numerator -, denominator +).

Standing waves are vital for musical instruments. In a string fixed at both ends or an open organ pipe (open at both ends), the fundamental frequency is $f_1 = v/2L$, and all integer multiples (harmonics) are possible ($f_n = n f_1$).

In a closed organ pipe (one end closed, one open), the fundamental frequency is $f_1 = v/4L$, and only odd harmonics are possible ($f_n = (2n-1)f_1$). Always pay attention to unit conversions (cm to m, kHz to Hz) and the specific conditions (open/closed pipe, moving towards/away) in numerical problems.

Prelims Revision Notes

1
Nature of Sound: — Mechanical, longitudinal wave. Requires a medium. Cannot travel in vacuum. Particles oscillate parallel to wave direction.
2
Wave Equation: — $v = flambda$. $v$: speed, $f$: frequency, $lambda$: wavelength.
3
Speed of Sound:

* Generally: $v_{\text{solids}} > v_{\text{liquids}} > v_{\text{gases}}$. * In gases: $v = sqrt{gamma P/\rho} = sqrt{gamma RT/M}$. * Effect of Temperature: $v propto sqrt{T}$ (absolute temperature). For air, $v_t = v_0 + 0.61t$ (approx. for $t$ in $^circ C$). * Effect of Humidity: Increases speed (humid air is less dense than dry air at same T, P). * Effect of Pressure: No effect on speed in gas if temperature is constant ($P/\rho$ remains constant).

1
Perception of Sound:

* Pitch: Determined by frequency ($f$). Higher $f implies$ higher pitch. Audible range: $20,\text{Hz}$ to $20,000,\text{Hz}$. * Loudness: Subjective perception of intensity. Determined by amplitude ($A$). $I propto A^2$. Intensity level $\beta = 10 log_{10}(I/I_0)$ in decibels (dB), where $I_0 = 10^{-12},\text{W/m}^2$. * Quality/Timbre: Determined by waveform (presence and relative intensity of overtones).

1
Wave Phenomena:

* Reflection: Echoes, reverberation. Laws of reflection hold. * Refraction: Bending of sound due to change in speed (e.g., temperature gradients). * Diffraction: Bending around obstacles/through openings. More pronounced for longer wavelengths (low frequencies). * Interference: Superposition of waves. Constructive (in phase, max amplitude), Destructive (out of phase, min amplitude).

1
Beats: — Formed by interference of two waves with slightly different frequencies ($f_1, f_2$). Beat frequency $f_{\text{beat}} = |f_1 - f_2|$. Used for tuning.
2
Doppler Effect: — Apparent change in frequency due to relative motion. f' = f left( \frac{v pm v_o}{v mp v_s} \right).

* Numerator: $v_o$ is observer speed. '+' if observer moves towards source, '-' if away. * Denominator: $v_s$ is source speed. '-' if source moves towards observer, '+' if away.

1
Standing Waves (Resonance):

* String fixed at both ends: Nodes at ends. $L = n(lambda_n/2)$. Frequencies $f_n = n(v/2L)$, where $n=1,2,3,dots$. All harmonics present. * Open Organ Pipe (open at both ends): Antinodes at ends.

$L = n(lambda_n/2)$. Frequencies $f_n = n(v/2L)$, where $n=1,2,3,dots$. All harmonics present. * Closed Organ Pipe (one end closed, one open): Node at closed end, antinode at open end. $L = (2n-1)(lambda_n/4)$.

Frequencies $f_n = (2n-1)(v/4L)$, where $n=1,2,3,dots$. Only odd harmonics present. * End Correction: For open end, effective length $L_{eff} = L + e$, where $e approx 0.6R$ (R=radius). For open pipe $L_{eff} = L + 2e$.

For closed pipe $L_{eff} = L + e$.

Vyyuha Quick Recall

For Doppler Effect sign convention: 'O'bserver 'T'owards 'S'ource, 'S'ource 'T'owards 'O'bserver.

Observer Towards Source: Observer gets a + (in numerator).
Source Towards Observer: Source gets a - (in denominator).

If away, just use the opposite sign. This helps remember which sign to use for $v_o$ and $v_s$ to get the correct frequency shift.