# Lecture 18: Mechanical Waves
## Physics I — Mechanics & Thermodynamics
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## Learning Objectives
By the end of this lecture, students will be able to:
1. Define mechanical waves and distinguish between transverse and longitudinal waves
2. Describe wave properties: wavelength, frequency, amplitude, speed, and phase
3. Write and interpret the sinusoidal wave equation
4. Calculate wave speed on a string from tension and linear mass density
5. Explain and apply the principle of superposition
6. Analyze standing waves on strings and in pipes and determine resonant frequencies
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## Lecture Content
### I. What Is a Mechanical Wave?
A **mechanical wave** is a disturbance that propagates through a medium, transporting energy without transporting matter. Three ingredients are required: a source of disturbance, a medium through which it can travel, and a physical mechanism to transmit the disturbance from one part of the medium to the next. The individual particles of the medium oscillate about their equilibrium positions but do not travel with the wave.
Waves come in two principal types. A **transverse wave** has particle displacement perpendicular to the direction of wave propagation. Examples include waves on a string and water surface waves (approximately). Electromagnetic waves are also transverse, though they require no medium. A **longitudinal wave** has particle displacement parallel to the direction of propagation. Sound waves and compression waves in a slinky are longitudinal. Some waves, such as surface water waves, have both transverse and longitudinal components.
### II. Wave Parameters
Several parameters characterize a wave. The **wavelength** (lambda) is the distance between two consecutive points in phase, such as crest to crest, measured in meters. The **period** (T) is the time for one complete cycle to pass a fixed point. The **frequency** (f) is f = 1/T, measured in Hz. The **angular frequency** is omega = 2 pi f, and the **wave number** is k = 2 pi / lambda, which can be thought of as a spatial frequency. The **amplitude** (A) is the maximum displacement from equilibrium. The **wave speed** is v = lambda f = omega / k. Crucially, the wave speed depends on the properties of the medium, not on the frequency or amplitude, which are determined by the source.
### III. The Sinusoidal Wave Equation
A sinusoidal transverse wave traveling in the +x direction is described by y(x, t) = A sin(kx - omega t + phi), where y is the displacement of the medium at position x and time t, and phi is the initial phase constant. For a wave traveling in the -x direction, the equation becomes y(x, t) = A sin(kx + omega t + phi).
At a fixed instant t, the graph of y versus x is a sinusoidal snapshot of the wave's spatial shape. At a fixed position x, the graph of y versus t shows the sinusoidal oscillation of that particular point in the medium. The particle velocity (distinct from the wave speed) is v_y = partial y / partial t = -A omega cos(kx - omega t), and the particle acceleration is a_y = -A omega^2 sin(kx - omega t) = -omega^2 y, confirming that each particle undergoes SHM.
Panel A: A snapshot of a transverse wave at time t = 0, showing y vs. x as a sine curve. Wavelength lambda is marked as the distance between two adjacent crests. Amplitude A is the maximum displacement from the x-axis. Panel B: The wave at a slightly later time t, shifted to the right by a distance v Delta t, illustrating the wave propagation. Panel C: A single particle at fixed position x shown oscillating up and down over time — the y vs. t graph is a sinusoid with period T.
### IV. Wave Speed on a String
The speed of a transverse wave on a stretched string is v = sqrt(F_T / mu), where F_T is the tension in the string and mu = m/L is the linear mass density (mass per unit length) in kg/m. Increasing the tension increases the wave speed, while increasing the mass density decreases it. The wave speed is **independent** of frequency and amplitude, both of which are set by the source rather than the medium.
### V. Energy Transport by Waves
Waves transport energy through the medium. The **power** transmitted by a sinusoidal wave on a string is P = (1/2) mu omega^2 A^2 v, proportional to the square of the amplitude, the square of the frequency, the wave speed, and the linear mass density. For waves spreading in three dimensions, the relevant quantity is **intensity** (I), defined as power per unit area: I = P/A_area, in W/m^2. For a point source radiating equally in all directions, I = P/(4 pi r^2), following an **inverse-square law**.
### VI. Superposition and Interference
The **principle of superposition** states that when two or more waves overlap, the resultant displacement is the algebraic sum of the individual displacements: y_total(x, t) = y_1(x, t) + y_2(x, t). **Constructive interference** occurs when waves are in phase, adding to produce a larger amplitude. This happens when the path difference is 0, lambda, 2 lambda, and so on. **Destructive interference** occurs when waves are out of phase, partially or fully canceling each other. This happens when the path difference is lambda/2, 3 lambda/2, and so on. Interference is a fundamental wave phenomenon that does not occur with particles.
### VII. Standing Waves
When two identical waves travel in opposite directions on a string, they produce a **standing wave** described by y(x, t) = 2A sin(kx) cos(omega t). **Nodes** are points that remain at zero displacement at all times, located at kx = n pi, or x = n lambda/2. **Antinodes** are points of maximum displacement, located halfway between adjacent nodes.
For a string fixed at both ends, the string length L must accommodate an integer number of half-wavelengths: L = n (lambda/2), where n = 1, 2, 3, ... The resonant frequencies, or harmonics, are f_n = n v / (2L) = n f_1. The first harmonic (n = 1) is the fundamental frequency, the second harmonic (n = 2) is the first overtone, and so on. The frequencies form a **harmonic series**: f_1, 2f_1, 3f_1, ...
Panel A: A string fixed at both ends showing the first three standing wave modes (n = 1, 2, 3). For each mode: the nodes (dots) and antinodes (maximum displacement points) are labeled, the wavelength is indicated (lambda_1 = 2L, lambda_2 = L, lambda_3 = 2L/3), and the frequency (f_1, 2f_1, 3f_1) is written. Panel B: A diagram showing two traveling waves (one moving right, one moving left) and their superposition at different instants, demonstrating how the standing wave pattern forms with stationary nodes.