Waves & Sound

Introduction to Wave Types and Wave Speed

Waves are disturbances that transfer energy through a medium (such as a string, water, or air) without transporting matter. They are classified based on the direction of particle displacement relative to wave motion.

• Transverse Waves: Displacement is perpendicular to wave motion (e.g., waves on a string).

• Longitudinal Waves: Displacement is parallel to wave motion (e.g., sound waves in air).

Key properties of waves:

• Wavelength (\( \lambda \)): Distance between consecutive crests (transverse) or compressions (longitudinal).

• Amplitude (A): Maximum displacement from equilibrium. For transverse waves, it is half the vertical distance from crest to trough.

• Period (T): Time to complete one cycle.

• Frequency (f): Number of cycles per second, \( f = \frac{1}{T} \).

All waves obey the speed relationship:

• \( v = \lambda f \)

Example: The wavelength of a sound wave with frequency 260 Hz and speed 343 m/s is \( \lambda = \frac{v}{f} = \frac{343}{260} = 1.32 \) m.

Wave Speed on a String

The speed of waves on a string depends on the string's tension, mass, and length. The wave speed is given by:

• \( v_{string} = \sqrt{\frac{F_T}{\mu}} \), where \( F_T \) is the tension and \( \mu = \frac{m}{L} \) is the mass per unit length.

Example: For a string with tension 100 N, mass 0.5 kg, and length 1.2 m, \( \mu = \frac{0.5}{1.2} = 0.417 \) kg/m. If \( \lambda = 0.15 \) m, \( v = \sqrt{\frac{100}{0.417}} = 15.5 \) m/s, and \( f = \frac{v}{\lambda} = \frac{15.5}{0.15} = 103 \) Hz.

Wave Speed in Fluids and Solids

For longitudinal waves, the speed depends on the medium's properties:

• In Fluids: \( v = \sqrt{\frac{\beta}{\rho}} \), where \( \beta \) is the bulk modulus and \( \rho \) is the density.

• In Solids: \( v = \sqrt{\frac{Y}{\rho}} \), where \( Y \) is Young's modulus.

Example: For a liquid with \( \rho = 1200 \) kg/m3, \( f = 400 \) Hz, \( \lambda = 8 \) m, \( v = 3200 \) m/s, so \( \beta = v^2 \rho = (3200)^2 \times 1200 = 1.23 \times 10^{10} \) Pa.

Wave Functions

A wave function describes the displacement \( y \) as a function of position \( x \) and time \( t \):

• \( y(x, t) = A \sin(kx \pm \omega t + \phi) \) or \( y(x, t) = A \cos(kx \pm \omega t + \phi) \)

• \( k = \frac{2\pi}{\lambda} \) (wavenumber), \( \omega = 2\pi f \) (angular frequency), \( \phi \) is the phase constant.

The sign in \( (kx \pm \omega t) \) depends on the direction of wave propagation.

Example: For a wave with amplitude 0.5 m, speed 8 m/s, and wavelength 0.32 m moving right, \( k = \frac{2\pi}{0.32} = 19.6 \) rad/m, \( \omega = 2\pi f = 2\pi \frac{8}{0.32} = 157 \) rad/s.

Transverse Velocity of Waves

The transverse velocity of a particle on the wave is the time derivative of the displacement:

• If \( y(x, t) = A \sin(kx \pm \omega t) \), then \( v_T(x, t) = \frac{\partial y}{\partial t} = \pm A \omega \cos(kx \pm \omega t) \).

• The maximum transverse velocity is \( v_{T, max} = A \omega \).

Wave Power and Intensity

Waves carry energy, and the rate of energy transfer is power. For waves on a string:

• Average power: \( P_{avg} = \frac{1}{2} \mu \omega^2 A^2 v \)

Wave intensity is power per unit area:

• \( I = \frac{P}{A} \)

• For spherical waves: \( I = \frac{P}{4\pi r^2} \)

Example: A loudspeaker radiating 500 W at 10 m: \( I = \frac{500}{4\pi (10)^2} = 0.398 \) W/m2.

Inverse-Square Law for Intensity

As distance from a point source increases, intensity decreases with the square of the distance:

• \( I_1/I_2 = (r_2/r_1)^2 \)

Example: If intensity is 0.25 W/m2 at 15 m, at what distance is intensity 0.01 W/m2? \( r_2 = r_1 \sqrt{I_1/I_2} = 15 \sqrt{0.25/0.01} = 75 \) m.

Sound Intensity Level (Decibels)

Sound intensity level is measured in decibels (dB):

• \( \beta = 10 \log_{10}\left(\frac{I}{I_0}\right) \), where \( I_0 = 1 \times 10^{-12} \) W/m2 (threshold of hearing).

Example: For \( I = 0.01 \) W/m2, \( \beta = 10 \log_{10}(0.01/1\times10^{-12}) = 10 \log_{10}(1\times10^{10}) = 100 \) dB.

Wave Interference and Superposition

When two or more waves meet, their displacements add (superposition principle):

• Constructive interference: Displacements have the same sign, amplitudes add.

• Destructive interference: Displacements have opposite signs, amplitudes subtract.

For sinusoidal waves: \( y_{net} = y_1 + y_2 \).

Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere. Only certain frequencies (harmonics) produce standing waves.

• Fundamental frequency (n=1): \( f_1 = \frac{v}{2L} \)

• nth harmonic: \( f_n = n f_1 = \frac{n v}{2L} \)

• Wavelength: \( \lambda_n = \frac{2L}{n} \)

Example: For a 1.5 m string, \( v = 48 \) m/s, \( f_1 = \frac{48}{3} = 16 \) Hz, \( \lambda_1 = 3 \) m.

Standing Sound Waves in Pipes

Standing waves can also form in pipes:

• Open at both ends: \( f_n = \frac{n v}{2L} \), \( \lambda_n = \frac{2L}{n} \), n = 1, 2, 3...

• Closed at one end: \( f_n = \frac{n v}{4L} \), \( \lambda_n = \frac{4L}{n} \), n = 1, 3, 5...

Example: For a 5 m open pipe, \( f_1 = \frac{343}{10} = 34.3 \) Hz.

Beats

Beats occur when two waves of slightly different frequencies interfere, causing oscillations in amplitude at the beat frequency:

• \( f_{beat} = |f_a - f_b| \)

• The perceived pitch is \( f_{sound} = \frac{f_a + f_b}{2} \)

Example: Two notes at 527 Hz and 524 Hz produce beats at 3 Hz.

The Doppler Effect

The Doppler Effect is the change in observed frequency due to relative motion between the source and the observer:

• \( f_L = f_S \frac{v + v_L}{v - v_S} \), where \( v \) is the speed of sound, \( v_L \) is the velocity of the listener (positive if moving toward the source), and \( v_S \) is the velocity of the source (positive if moving away from the listener).

Example: If a stationary car alarm emits 550 Hz and a listener moves toward it and hears 600 Hz, solve for the listener's speed using the Doppler formula.

Summary Table: Key Wave Equations

Summary Table: Standing Waves in Strings and Pipes

Additional info:

• For all wave equations, ensure units are consistent (e.g., SI units).

• For the Doppler Effect, the sign convention is crucial: positive velocities are toward each other.

• In standing waves, "harmonic" refers to the integer n, while "overtone" is the number of frequencies above the fundamental (nth overtone = (n+1)th harmonic).