12 - Mechanical Waves and Sound

A structured guide to mechanical waves, their mathematical relationships, interference and resonance, standing-wave patterns, sound, and energy transfer.

Wave Fundamentals

A is a traveling disturbance that requires a material medium. The medium may be a string, water, air, or a solid. As the disturbance propagates, particles oscillate about equilibrium positions rather than traveling with the wave over long distances. The wave transfers energy and momentum, while particle motion is generally local.

For a periodic wave, the main quantities are:

, $A$ : maximum displacement from equilibrium.
, $\lambda$ : distance between corresponding points on successive cycles.
Period, $T$ : time for one complete cycle.
, $f$ : number of cycles per second.
, $v$ : speed of propagation through the medium.
Phase: position within a cycle of oscillation.

and period are related by

f=\frac{1}{T}.

, , , and period are related by

v=f\lambda=\frac{\lambda}{T}.

The speed is determined mainly by the medium and its conditions. For example, a wave traveling along a rope at $12\ \text{m/s}$ with $3.0\ \text{Hz}$ has

\lambda=\frac{v}{f}=\frac{12}{3.0}=4.0\ \text{m}.

Takeaway: A wave carries energy through a medium, and the basic relationship $v=f\lambda$ connects its speed, , and .

Wave Types and Mathematical Models

The direction of particle motion relative to propagation determines the wave type.

In a , particles oscillate perpendicular to the direction of travel. Such waves have crests and troughs, and the distance from equilibrium to a crest or trough is the . Waves on stretched strings are a common example.

In a , particles oscillate parallel to the direction of travel. The pattern contains alternating compressions, where particles are closer together and pressure is relatively high, and rarefactions, where particles are farther apart and pressure is relatively low.

A sinusoidal wave traveling in the positive $x$ -direction can be represented by

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

where

k=\frac{2\pi}{\lambda}, \qquad \omega=2\pi f.

The phase constant is $\phi$ . The expression $kx-\omega t$ indicates motion in the positive direction; using $kx+\omega t$ indicates motion in the negative direction. The can also be written as

v=\frac{\omega}{k}.

For a stretched string, the speed depends on tension and linear mass density:

v=\sqrt{\frac{F_T}{\mu}}, \qquad \mu=\frac{m}{L}.

Increasing tension increases , while increasing mass per unit length decreases it.

Takeaway: Classify a wave by comparing particle motion with propagation, then choose an equation that matches the medium and the wave pattern.

Superposition and Interference

When waves overlap, the applies: the resultant displacement is the algebraic sum of the individual displacements.

y_{\text{resultant}}=y_1+y_2+y_3+\cdots

occurs when disturbances reinforce one another. For two waves in phase,

A_{\text{resultant}}=A_1+A_2.

occurs when disturbances oppose one another. For two waves,

A_{\text{resultant}}=|A_1-A_2|.

Equal amplitudes with opposite displacements can produce zero resultant displacement at a location. This does not violate conservation of energy; energy is redistributed and may appear elsewhere or in another form.

occur when two waves have slightly different frequencies. Their combined alternately increases and decreases, with beat

f_{\text{beat}}=|f_1-f_2|.

For frequencies of $256\ \text{Hz}$ and $260\ \text{Hz}$ ,

f_{\text{beat}}=|260-256|=4\ \text{Hz}.

Takeaway: Interference changes the temporary resultant displacement, while the component waves continue to propagate.

Standing Waves and Boundary Conditions

A forms when two waves with the same , , and travel in opposite directions and superpose. Reflection from a boundary commonly creates this situation. The pattern contains , which remain at rest, and , which oscillate with maximum .

The spacing rules are

\text{node to adjacent node}=\frac{\lambda}{2}, \qquad \text{antinode to adjacent antinode}=\frac{\lambda}{2}, \qquad \text{node to nearest antinode}=\frac{\lambda}{4}.

For a string fixed at both ends, both endpoints must be . The allowed wavelengths and frequencies are

L=n\frac{\lambda_n}{2}, \qquad \lambda_n=\frac{2L}{n}, \qquad n=1,2,3,\ldots

and

f_n=\frac{nv}{2L}.

The case $n=1$ is the fundamental mode, or first harmonic. Higher values are higher harmonics or normal modes. For a string with $L=1.2\ \text{m}$ and $v=60\ \text{m/s}$ ,

f_1=\frac{60}{2(1.2)}=25\ \text{Hz}, \qquad f_2=2f_1=50\ \text{Hz}.

An ideal does not carry net energy steadily from one end to the other; energy oscillates between kinetic and potential forms within the pattern.

Takeaway: Boundary conditions determine the allowed standing-wave patterns and frequencies.

in Strings and Air Columns

occurs when a periodic driving force has a equal or close to a system's natural . The driving force then adds energy efficiently during each cycle, producing a large . A small force can create a large response when timing is appropriate and damping is small.

Standing waves are a form of because only frequencies that satisfy the system's boundary conditions can persist. For a string fixed at both ends, the resonant frequencies are

f_n=\frac{nv}{2L}.

Damping removes mechanical energy and limits the resonant . Examples include pushing a swing at the correct timing, a string vibrating at a natural , air resonating in a pipe, and a structure responding strongly to periodic ground motion.

For air columns:

A tube open at both ends has displacement at both ends and allowed frequencies $f_n=\frac{nv}{2L}$ , where $n=1,2,3,\ldots$ .
A tube closed at one end and open at the other has a displacement node at the closed end and an antinode at the open end. Only odd harmonics occur:

\lambda_n=\frac{4L}{n}, \qquad f_n=\frac{nv}{4L}, \qquad n=1,3,5,\ldots

The fundamental for a tube closed at one end is $\lambda_1=4L$ .

Takeaway: is controlled by driving , natural , damping, and boundary conditions.

and Mechanical-Wave Energy

is a mechanical disturbance that propagates through a medium. In air, it is a longitudinal variation in pressure and density: compressions have higher pressure, and rarefactions have lower pressure. cannot travel through a vacuum because a vacuum has no particles to transmit the disturbance.

generally travels faster in solids than in liquids and faster in liquids than in gases. Near room temperature, its speed in air is approximately

v_{\text{sound}}\approx343\ \text{m/s}

at $20^\circ\text{C}$ .

affects perceived pitch, while is related to perceived loudness. The medium primarily determines , so a louder does not necessarily travel faster. is the rate at which wave energy passes through a unit area:

I=\frac{P}{A}.

For a spherical wave spreading uniformly from a point source,

I=\frac{P}{4\pi r^2}.

Thus, decreases with the inverse square of distance. Doubling the distance reduces to one-fourth. For many mechanical waves, is proportional to the square of :

I\propto A^2.

Doubling can therefore increase transported energy rate by a factor of four, provided the medium remains in the linear regime.

Takeaway: transfers energy through local particle oscillations; controls pitch, relates to loudness, and distance affects .

Wave-Problem Strategy

A reliable solution begins by identifying the physical model rather than immediately substituting numbers.

Decide whether the disturbance is transverse, longitudinal, or a combination.
List the known quantities, such as $v$ , $f$ , $T$ , $\lambda$ , $A$ , $L$ , or $\mu$ .
Select the appropriate relationship:
- Use $v=f\lambda$ for a periodic traveling wave.
- Use $v=\sqrt{F_T/\mu}$ for a wave on a stretched string.
- Use standing-wave boundary conditions for strings and air columns.
- Use $f_{\text{beat}}=|f_1-f_2|$ for .
- Use $I=P/A$ or $I=P/(4\pi r^2)$ for energy spreading.
Draw the wave pattern when , , endpoints, or open and closed boundaries matter.
Check units: in hertz, and length in meters, and speed in meters per second.
Interpret the result physically. In a fixed medium, does not normally determine ; the medium does.

Takeaway: A correct model, a clear boundary-condition diagram, and a unit check prevent most wave-problem errors.