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Topic 17 of 20

Mechanical Waves

Mechanical waves transfer energy through a medium via oscillations, without any net transport of matter. Understanding waves — their speed, superposition, interference, and resonance — is foundational for understanding sound, light, quantum mechanics, and almost every branch of physics.

Key Concepts

Wave Parameters

A wave is characterized by wavelength

\lambda

(spatial repeat length), frequency

f

(oscillations per second), amplitude

A

, and wave speed

v = \lambda f

. Angular wave number

k = 2\pi/\lambda

Transverse vs. Longitudinal

Transverse: medium oscillates perpendicular to propagation (e.g., string waves, EM waves). Longitudinal: oscillation is parallel to propagation (e.g., sound, compression waves in springs).

Wave Equation

A sinusoidal wave traveling in the +x direction:

y(x,t) = A\sin(kx - \omega t + \phi)

. The phase

(kx - \omega t)

moves at speed

v = \omega/k = \lambda f

Wave Speed on a String

v = \sqrt{T/\mu}

where

T

is the tension and

\mu = m/L

is the linear mass density (kg/m). Stiffer/lighter string → faster wave.

Superposition and Interference

When two waves overlap, their displacements add algebraically. Constructive interference: waves in phase, amplitudes add. Destructive interference: waves 180° out of phase, amplitudes cancel.

Standing Waves

Superposition of identical waves traveling in opposite directions. Fixed points (nodes) and maximum-displacement points (antinodes) do not move. Standing waves form only at resonant frequencies.

Key Equations

Wave speed

v = \lambda f = \frac{\omega}{k}

Fundamental relation between speed, wavelength, and frequency.

Wave on a string

v = \sqrt{\frac{T}{\mu}}

T = tension (N), μ = linear mass density (kg/m). Speed independent of frequency.

Traveling wave

y(x,t) = A\sin(kx - \omega t), \quad k = \frac{2\pi}{\lambda}

Snapshot at t=0: sinusoid in x. At fixed x: sinusoidal oscillation in time.

Resonant frequencies (string, both ends fixed)

f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}} = \frac{nv}{2L}, \quad n = 1,2,3,\ldots

n = 1 is the fundamental (first harmonic); n = 2 is the second harmonic, etc.

Wave intensity

I \propto A^2, \quad I = \frac{P}{4\pi r^2} \text{ (point source)}

Intensity (power per unit area) falls as 1/r² from a point source in 3D.

Worked Example

Standing Waves on a Guitar String

Problem

A guitar string of length 0.65 m has linear density $\mu = 3\times10^{-3}$ kg/m and tension $T = 75$ N. Find the fundamental frequency and the second harmonic.

Solution

Wave speed on the string:

v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{75}{3\times10^{-3}}} = \sqrt{25000} = 158 \text{ m/s}

Fundamental ( $n=1$ ):

f_1 = \frac{v}{2L} = \frac{158}{2\times0.65} \approx 122 \text{ Hz}

Second harmonic ( $n=2$ ): $f_2 = 2f_1 \approx 244$ Hz.

Answer Fundamental ≈ 122 Hz; second harmonic ≈ 244 Hz.

Practice

Exercises

7 problems

1 of 7

A wave on a string has frequency $f = 50$ Hz and wavelength $\lambda = 0.6$ m. What is the wave speed (in m/s)?

Your answer

m/s

2 of 7

A string of mass $m = 5$ g and length $L = 2$ m is under tension $T = 80$ N. What is the wave speed (in m/s) on the string?

Your answer

m/s

3 of 7

What is the fundamental frequency (in Hz) of a string of length $L = 0.8$ m with wave speed $v = 240$ m/s?

Your answer

4 of 7

What is the third harmonic frequency (in Hz) of the string in Exercise 3?

Your answer

5 of 7

A sound point source radiates $P = 10$ W uniformly. What is the intensity (in W/m²) at a distance of $r = 5$ m?

Your answer

W/m²

6 of 7

Two waves on the same string have amplitudes $A_1 = 3$ cm and $A_2 = 3$ cm and are exactly in phase. What is the amplitude (in cm) of the superposed wave?

Your answer

7 of 7

A standing wave on a $1.5$ m string fixed at both ends has 4 antinodes. What is the wavelength (in m) of the wave?

Your answer

Key Takeaways

Wave speed $v = \lambda f$ ; for a string, $v = \sqrt{T/\mu}$ — speed is a property of the medium, not the frequency.
Transverse waves (perpendicular oscillation) vs. longitudinal (parallel oscillation); sound is longitudinal.
Standing waves form when the string length accommodates a whole number of half-wavelengths: $L = n\lambda/2$ .
Intensity is proportional to amplitude squared; it falls as $1/r^2$ from a point source.
Constructive interference when path difference = $n\lambda$ ; destructive when path difference = $(n+\frac{1}{2})\lambda$ .