Sound Waves – Properties, Speed, Equations & Doppler Effect | Entrance Exam Physics Notes

SOUND WAVES — SHORT NOTES (Entrance Exam Focus)

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1. Nature of Sound Waves

• Sound is a mechanical, longitudinal wave.

• Travels through compression and rarefaction.

• Requires a medium (cannot travel in vacuum).

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2. Speed of Sound (v)

General formula

$$v=\sqrt{\frac{\gamma P}{\rho }}$$

In Solids

$$v=\sqrt{\frac{Y}{\rho }}$$

(Y = Young’s modulus)

In Liquids

$$v=\sqrt{\frac{B}{\rho }}$$

(B = Bulk modulus)

In Gases

$$v=\sqrt{\gamma \frac{RT}{M}}$$

Temperature dependence (most important)

$$v=331+0.6T(\text{m/s})$$

(T in °C)

Speed of sound is highest in solids, then liquids, lowest in gases.

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3. Displacement, Pressure & Velocity Curves

• Sound waves: longitudinal

• Pressure & density variations are 90° out of phase with particle displacement.

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4. Intensity & Loudness

Intensity (I)

$$I=\frac{P_{wave}}{\mathrm{A<span\; class="vlist-s"></span>}}$$

Intensity Level (in decibel)

$$\beta =10\log \left(\frac{I}{I_0}\right)$$

Where $I_0={10}^{-12}{\text{W/m}}^2$

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5. Beats

When two waves of slightly different frequencies interfere:

$$f_{beat}=\mathrm{\mid }{f}_1-f_2\mathrm{\mid }$$

Beats used to tune musical instruments.

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6. Doppler Effect (VERY IMPORTANT)

General formula

$$f^{\mathrm{\prime }}=f\left(\frac{v\pm v_o}{v\mp v_s}\right)$$

• vo: velocity of observer

• vs: velocity of source

Use PLUS when motion reduces distance.
Use MINUS when motion increases distance.

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7. Reflection – Echo

Echo heard if sound returns after 0.1 s.

Minimum distance:

$$d=\frac{v\times t}{2}\approx 17\text{ m}$$

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8. Reverberation (Acoustics)

Persistence of sound due to repeated reflections.

Good hall acoustics → optimum reverberation.

Sabine’s Formula:

$$T=0.161\frac{V}{A}$$

(V) = volume, (A) = total absorption.

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9. Interference of Sound

Constructive:

$$\mathrm{\Delta }\varphi =2n\pi$$

Destructive:

$$\mathrm{\Delta }\varphi =(2n+1)\pi$$

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10. Stationary Wave in Air Columns

Open pipe

$$f_n=\frac{nv}{2\mathrm{L}}$$

Closed pipe

$$f_n=\frac{nv}{4L}\text{(n = odd only)}$$

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11. Resonance Tube

Resonance occurs when:

$$L=(n-\frac{1}{4})\lambda$$ (n = 1, 3, 5…)

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12. High-Yield Numerical Topics

• Speed of sound at different temperatures

• Beats calculation

• Doppler shift

• Fundamental & harmonic frequencies of air columns

• Power and intensity level

• Echo time & distance

• Superposition & interference

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