Transverse and Longitudinal Waves

Fig. 14.2 is a simple but essential sketch that establishes the geometry of a wave on a string. The string is stretched horizontally between a hand on the left and a fixed point on a wall at the right. The hand has just given a quick upward flick, creating a single hump — a pulse — that is already moving to the right. The pulse is drawn as a smooth bump above the string’s equilibrium line, and an arrow above it labels the direction of travel (the $x$-direction). The axes are placed at the bottom-left: the $x$-axis runs horizontally along the string’s rest position, and the $y$-axis points vertically upward.

What the figure drives home is the distinction between the direction of the wave’s motion and the direction in which the string’s particles move. The pulse travels along the $x$-axis, but each small element of the string (a tiny segment) only moves up and down — that is, along the $y$-axis. No piece of string moves horizontally with the pulse; they simply rise and then fall back as the disturbance passes. This is the defining feature of a transverse wave: the particle displacement is perpendicular to the wave velocity.

The textbook uses this figure to introduce the wave function $y(x,t)$, which describes the vertical displacement of a string element at position $x$ at time $t$. For a pulse that keeps its shape as it moves, the wave function takes the form

for a pulse travelling to the right with speed $v$, or $y(x,t) = f(x + vt)$ for one travelling left. Here $f$ is the shape of the pulse at $t=0$. The argument $x - vt$ means that the entire shape shifts rightward by $vt$ after time $t$ — the pulse moves, but the string elements only oscillate vertically.

The figure also sets up the coordinate system used throughout the chapter. The $x$-axis is the undisturbed string line, and $y$ measures the displacement from that line. The hand at the left end is the source of the disturbance; the fixed wall at the right is a boundary where the pulse will later reflect. In later sections, the same geometry is used to derive the wave equation for a stretched string:

where $v = \sqrt{T/\mu}$, with $T$ the tension in the string and $\mu$ its mass per unit length. That equation emerges directly from applying Newton’s second law to a small segment of the string — the same segment that Fig. 14.2 shows oscillating up and down. …

Fig. 14.3 is a snapshot of a sinusoidal wave on a string at one instant of time. The string is stretched horizontally from a hand on the left to a fixed wall (shown hatched) on the right. A horizontal arrow labelled "Harmonic wave" points from left to right, indicating the direction in which the wave pattern moves. Two axes are drawn: the $x$-axis runs along the string's equilibrium (undisturbed) position, and the $y$-axis points perpendicular to it, measuring the displacement of each point on the string from its rest position.

The curve itself is a smooth, continuous sine wave. Each small element of the string oscillates up and down (along $y$) while the wave travels rightward (along $x$). This is the defining feature of a transverse wave: the motion of the medium is perpendicular to the direction of energy transfer. The figure makes this distinction visually — the hand at left provides the disturbance, the wall at right is a fixed end, and the wave shape moves between them without any net motion of the string itself to the right.

The textbook uses this figure to introduce the mathematical description of a harmonic wave. For a wave travelling in the positive $x$-direction, the displacement $y$ of a point at position $x$ at time $t$ is given by

where:

• $A$ is the amplitude — the maximum displacement from equilibrium (the peak height of the curve).
• $k = \frac{2\pi}{\lambda}$ is the angular wave number, with $\lambda$ the wavelength (the distance between two successive crests or troughs).
• $\omega = \frac{2\pi}{T}$ is the angular frequency, with $T$ the time period of oscillation.
• $\phi$ is the initial phase (often set to zero for simplicity).

The argument $(kx - \omega t + \phi)$ is called the phase of the wave. The minus sign indicates propagation to the right; a plus sign would give a wave moving left. The wave speed $v$ is related to these parameters by $v = \frac{\omega}{k} = \lambda f$, where $f = 1/T$ is the frequency.

Fig. 14.4 is a schematic of a vertical pipe, open at the top and closed at the bottom by a movable piston. The pipe is filled with air, shown as a stippled region. The piston has a circular handle and is marked with an up-down arrow, indicating it oscillates vertically. Inside the pipe, the stippling is not uniform: there are alternating bands of denser stippling (compressions) and sparser stippling (rarefactions). The entire assembly is labelled "Air".

The figure teaches the core idea of a longitudinal wave. As the piston moves up and down, it alternately pushes the air layer next to it (compression) and pulls it apart (rarefaction). This disturbance travels along the pipe, not as a sideways wiggle but as a sequence of crowded and spread-out regions. The volume elements of air — small imaginary chunks — oscillate back and forth parallel to the direction the wave travels (vertically). That is the defining feature of a longitudinal wave: the particle displacement is along the line of propagation, not perpendicular to it.

The textbook uses this figure to introduce the mathematical description of a longitudinal wave. For a wave travelling in the $+x$ direction, the displacement $y(x,t)$ of a particle from its equilibrium position at time $t$ is given by the same sinusoidal form used for transverse waves:

Here:

• $y$ is the instantaneous displacement of a particle from its mean position (positive means forward along $x$, negative means backward).
• $A$ is the amplitude — the maximum displacement of any particle.
• $\omega = 2\pi f$ is the angular frequency, where $f$ is the frequency of the piston's oscillation.
• $k = 2\pi / \lambda$ is the angular wavenumber, where $\lambda$ is the wavelength — the distance between two successive compressions (or rarefactions).
• $\phi$ is the initial phase, set by the piston's position at $t=0$.

The figure also leads to the relation between the wave speed $v$, frequency $f$, and wavelength $\lambda$:

$v = f \lambda$ …