# Sound Wave Formulas

Sound waves in simple terms are longitudinal waves that travel through a medium like air or water. In this article, we will see some important sound wave formulas which will also help you in preparation for competitive exams like JEE and NEET.

Index

## Longitudinal displacement of sound wave

$$\xi = A Sin ( \omega t – kx)$$

## Pressure excess during traveling sound wave

$$P_{ex} = -B \frac{\delta \xi}{\delta x}$$ $$= (BAk) \,cos\, (\omega t – kx)$$

(it is true for travelling wave as well as standing waves)

The amplitude of pressure excess = BAk

## Speed of sound

C = $$\sqrt \frac{E}{\rho}$$

Where E = Elastic modulus for the medium

$$\rho$$ = density of medium

• for solid $$\quad C = \sqrt \frac{Y}{\rho}$$, where Y = young’s modulus for the solid
• for liquid $$\quad C = \sqrt \frac{B}{\rho}$$, where B = Bulk modulus for the liquid
• for gases $$\quad C = \sqrt \frac{B}{\rho} = \sqrt \frac{\gamma P}{\rho} = \sqrt \frac{\gamma RT}{M_{0}}$$, where $$M_{0}$$ is molecular wt. of the gas in (kg/mole)

Intensity of sound wave:

$$I = 2 \pi^{2} f^2 A^2 \rho v = \frac{P^2_{m}}{2 \rho v}$$ where, $$I \propto P_{m}^2$$

## Loudness of sound

$$L = 10 \, log_{10} [\frac{I}{I_{0}}]dB$$

where $$I_{0} = 10^{-12} W/m^{2}$$ (This the minimum intensity human ears can
listen)

Intensity at a distance r from a point source $$= I = \frac{P}{4 \pi r^2}$$

## Interference of Sound Wave

if $$p_{1} = p_{m1} sin (\omega t – kx_{1}+ \theta_{1})$$

if $$p_{2} = p_{m2} sin (\omega t – kx_{2}+ \theta_{2})$$

resultant excess pressure at point O is $$= P_{1}+ P_{2}$$

$$p = p_{0} sin (\omega t – kx+ \theta )$$

$$p_{0}= \sqrt p^2_{m1} + p^2_{m2}+2p_{m1} \, p_{m}2 cos \phi$$

where $$\phi = [k (x_{2}- x_{1})+ (\theta _{1}+\theta _{2})]$$

and $$I = I_{1} + I_{2} + 2 \sqrt I_{1} I_{2}$$

1. For constructive interference

$$\phi = 2n \pi$$ and $$\Rightarrow p_{0} = p_{m1}+p_{m2}$$ constructive interference

2. For destructive interference

$$\phi = (2n +1) \pi$$ and $$\Rightarrow p_{0} = |p_{m1}+p_{m2}|$$ destructive interference)

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 constructive interference : $$\Delta X = (2n + 1) \frac{\lambda }{2}$$

a) if $$\quad p_{m1}= p_{m2}$$ and $$\theta = \pi ,\, 3 \pi ,\, \cdots$$

resultant p = 0 i.e. no sound

b) if $$\quad p_{m1}= p_{m2}$$ and $$\phi = 0 , \,2 \pi ,\, 4 \pi ,\, \cdots$$

$$p_{0} = 2 p_{m}$$ and $$I_{o} = 4 I_{1}$$

$$p_{0} = 2 p_{m1}$$

## Close Organ pipe

$$f = \frac{v}{4 \ell}+\frac{3v}{4 \ell}+\frac{5v}{4 \ell} , \cdots \frac{nV}{2 \ell}$$

## Open Organ pipe

$$f = \frac{v}{2 \ell}+\frac{2v}{2 \ell}+\frac{3v}{2 \ell} , \cdots \frac{nV}{2 \ell}$$

Beats : Beats frequency = $$|f_{1}-f_{2}|$$

## Doppler’s Effect

The observed frequency, $$f’ = [\frac{v-v_{0}}{v- v_{s}}]$$

and Apparent wavelength $$\lambda ‘ = \lambda [\frac{v- v_{s}}{v}]$$

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