Longitudinal Displacement of a Sound Wave:

The longitudinal displacement of a sound wave can be expressed as: $$ \xi = A \sin(\omega t - kx) $$ where:

• $\xi$ is the displacement,
• $A$ is the amplitude of the wave,
• $\omega$ is the angular frequency,
• $t$ is the time,
• $k$ is the wave number, and
• $x$ is the position.

Pressure Excess During a Travelling Sound Wave:

The pressure excess in a travelling sound wave can be expressed as: $$ P_{\text{ex}} = -B \frac{\partial \xi}{\partial x} = (BAk) \cos(\omega t - kx) $$ where:

• $P_{\text{ex}}$ is the pressure excess,
• $B$ is the bulk modulus of the medium,
• $A$ is the amplitude of the wave,
• $k$ is the wave number,
• $\omega$ is the angular frequency,
• $t$ is the time, and
• $x$ is the position.

The amplitude of the pressure excess is given by $BAk$.

Speed of Sound:

The speed of sound in a medium can be expressed as: $$ C = \sqrt{\frac{E}{\rho}} $$ where:

• $C$ is the speed of sound,
• $E$ is the elastic modulus of the medium, and
• $\rho$ is the density of the medium.

For different types of media, the elastic modulus is represented as follows:

• For solids: $E = Y$, where $Y$ is Young’s modulus.
• For liquids: $E = B$, where $B$ is the bulk modulus.
• For gases: $E = B = \gamma P = \gamma \frac{RT}{M_0}$, where $\gamma$ is the adiabatic index, $P$ is the pressure, $R$ is the universal gas constant, $T$ is the temperature, and $M_0$ is the molar mass.

Intensity of a Sound Wave:

The average intensity of a sound wave can be expressed as: $$ \langle I \rangle = 2\pi^2 f^2 A^2 \rho v = \frac{P_m^2}{2\rho v} $$ where:

• $\langle I \rangle$ is the average intensity,
• $f$ is the frequency of the wave,
• $A$ is the amplitude of the wave,
• $\rho$ is the density of the medium,
• $v$ is the speed of sound in the medium, and
• $P_m$ is the maximum pressure.

The intensity is proportional to the square of the maximum pressure, i.e., $\langle I \rangle \propto P_m^2$.

Loudness of Sound:

The loudness of sound, measured in decibels (dB), is given by: $$ L = 10 \log_{10}\left(\frac{I}{I_0}\right) \text{ dB} $$ where:

• $L$ is the loudness,
• $I$ is the intensity of the sound, and
• $I_0 = 10^{-12} \text{ W/m}^2$ is the reference intensity, which is the minimum intensity detectable by the human ear.

The intensity at a distance $r$ from a point source is given by: $$ I = \frac{P}{4\pi r^2} $$ where $P$ is the power of the source.

Interference of Sound Waves:

When two sound waves interfere, the resultant excess pressure at a point $O$ is given by: $$ p = P_1 + P_2 = p_0 \sin(\omega t - kx + \theta) $$ where:

• $P_1 = p_{m1} \sin(\omega t - kx_1 + \theta_1)$

• $P_2 = p_{m2} \sin(\omega t - kx_2 + \theta_2)$

• $p_0 = \sqrt{p_{m1}^2 + p_{m2}^2 + 2p_{m1}p_{m2}\cos\phi}$

• $\phi = k(x_2 - x_1) + (\theta_1 - \theta_2)$

• $I = I_1 + I_2 + 2\sqrt{I_1 I_2}$

For constructive interference:

• $\phi = 2n\pi$

• $p_0 = p_{m1} + p_{m2}$

For destructive interference:

• $\phi = (2n + 1)\pi$

• $p_0 = |p_{m1} - p_{m2}|$

If $\phi$ is due to path difference only, then $$\phi = \frac{2 \pi}{\lambda} \Delta x$$

Condition for constructive interference: $$\Delta x = n \lambda$$

Condition for destructive interference: $$\Delta x = (2n + 1) \frac{\lambda}{2}$$

Closed Organ Pipe:

For a closed organ pipe, the frequencies of the harmonics are given by: $$ f = \frac{v}{4\ell}, \frac{3v}{4\ell}, \frac{5v}{4\ell}, \ldots, \frac{(2n+1)v}{4\ell} $$ where $n$ is the overtone number.

Open Organ Pipe:

For an open organ pipe, the frequencies of the harmonics are given by: $$ f = \frac{v}{2\ell}, \frac{2v}{2\ell}, \frac{3v}{2\ell}, \ldots, \frac{nv}{2\ell} $$

Beats:

The frequency of beats is given by: $$ \text{Beat frequency} = |f_1 - f_2| $$

Doppler’s Effect:

The observed frequency and apparent wavelength in Doppler’s effect are given by: $$ f’ = f\left(\frac{v - v_0}{v - v_s}\right) $$ $$ \lambda’ = \lambda\left(\frac{v - v_s}{v}\right) $$ where:

• $f’$ is the observed frequency,
• $f$ is the source frequency,
• $v$ is the speed of sound in the medium,
• $v_0$ is the speed of the observer,
• $v_s$ is the speed of the source,
• $\lambda’$ is the apparent wavelength, and
• $\lambda$ is the source wavelength.