16-1 Transverse Waves#

Prompts

What are the three main types of waves? Which require a material medium?
Distinguish transverse and longitudinal waves. For a wave on a string, which way do the string elements move relative to the wave’s direction of travel?
Write the displacement \(y(x,t)\) for a sinusoidal wave traveling in the \(+x\) direction. Identify amplitude \(y_m\), angular wave number \(k\), angular frequency \(\omega\), and phase. How are \(k\), \(\lambda\), \(\omega\), \(T\), and \(f\) related?
A wave has \(y(x,t) = y_m \sin(kx - \omega t)\). What is the wave speed \(v\)? How do you tell if a wave travels in the \(+x\) or \(-x\) direction from its equation?
For a string element at fixed \(x\), how does its transverse velocity \(u = \partial y/\partial t\) vary with time? When is \(u\) maximum? When is it zero?

Lecture Notes#

Overview#

Waves propagate disturbances through a medium (or vacuum for EM waves). The wave form moves; the medium oscillates in place.
Transverse wave: particles oscillate perpendicular to the direction of travel (e.g., string waves).
Longitudinal wave: particles oscillate parallel to the direction of travel (e.g., sound in air).
A sinusoidal wave has the form \(y(x,t) = y_m \sin(kx - \omega t + \phi)\); its speed is \(v = \omega/k = \lambda f\).

Transverse	Longitudinal
Displacement \(\perp\) propagation	Displacement \(\parallel\) propagation
String, EM waves	Sound, slinky compression

Types of waves#

Mechanical: require a material medium (string, water, air); governed by Newton’s laws.
Electromagnetic: no medium needed; travel at \(c\) in vacuum.
Matter waves: associated with particles (quantum).

This chapter focuses on mechanical waves on a string.

Transverse vs longitudinal#

Transverse (e.g., string): Pluck the string; a pulse travels along it. Each string element moves up and down while the pulse moves horizontally. The displacement is perpendicular to the wave velocity.

Longitudinal (e.g., sound in a pipe): A piston compresses air; the compression travels along the pipe. Air elements move back and forth along the direction of travel.

Wave vs medium

The wave moves; the medium does not. A string element oscillates about its equilibrium position. The wave form (the pattern) travels from one end to the other. Same for sound: air molecules oscillate; the compression wave propagates.

Sinusoidal wave: displacement function#

A sinusoidal wave traveling in the positive \(x\) direction:

(90)#\[ y(x,t) = y_m \sin(kx - \omega t + \phi) \]

Symbol	Name	Meaning
\(y_m\)	Amplitude	Maximum displacement (always positive)
\(k\)	Angular wave number	\(k = 2\pi/\lambda\) (rad/m)
\(\omega\)	Angular frequency	\(\omega = 2\pi/T = 2\pi f\) (rad/s)
\(kx - \omega t + \phi\)	Phase	Argument of the sine
\(\phi\)	Phase constant	Sets \(y\) and slope at \(x=0\), \(t=0\)

Wavelength \(\lambda\): distance over which the pattern repeats. Period \(T\): time for one full oscillation of a string element.

Wave speed and direction#

For a point of constant phase: \(kx - \omega t = \text{const}\) \(\Rightarrow\) \(k\,dx/dt - \omega = 0\), so

(91)#\[ v = \frac{dx}{dt} = \frac{\omega}{k} = \frac{\lambda}{T} = \lambda f \]

Direction of travel:

\(y = h(kx - \omega t)\): wave travels in \(+x\) direction.
\(y = h(kx + \omega t)\): wave travels in \(-x\) direction.

Important

Any function of the form \(y(x,t) = h(kx \pm \omega t)\) represents a traveling wave with speed \(v = \omega/k\). The shape is given by \(h\); the \(\pm\) fixes the direction.

Poll: Snapshot to equation

Given a snapshot at \(t = 0\) of a transverse wave traveling right with \(v = 2\) m/s and \(\lambda = 4\) m, which equation is correct?

[FIGURE: Snapshot of the transverse wave in space, showing amplitude and wavelength]

(A) \(y = 2\sin[(\pi/2)x - \pi t]\)
(B) \(y = 2\sin[\pi x - (\pi/2)t]\)
(C) \(y = 2\sin[(\pi/2)x - (\pi/2)t]\)
(D) \(y = 2\sin[\pi x - \pi t]\)
(E) None of these

Transverse velocity and acceleration#

For a string element at fixed \(x\), \(y\) varies with \(t\)—the element undergoes SHM. Its transverse velocity and acceleration:

(92)#\[ u = \frac{\partial y}{\partial t} = -\omega y_m \cos(kx - \omega t + \phi) \]

(93)#\[ a_y = \frac{\partial u}{\partial t} = -\omega^2 y_m \sin(kx - \omega t + \phi) = -\omega^2 y \]

\(u\) maximum when \(y = 0\) (element at equilibrium, moving fastest).
\(u = 0\) when \(y = \pm y_m\) (element at extremes, momentarily at rest).

Caution

Do not confuse \(v\) (wave speed, along \(x\)) with \(u\) (transverse velocity of a string element, along \(y\)).

Poll: Phase constant from graph

A plot of \(f(\theta)\) is a sine wave shifted horizontally. Which best describes it?

[FIGURE: Sine wave shifted horizontally; show 2–3 cycles with labeled axes]

(A) \(\sin(\theta + \pi/4)\)
(B) \(\sin(\theta - \pi/4)\)
(C) \(\sin(\theta + 3\pi/4)\)
(D) \(\sin(\theta - 3\pi/4)\)

Example: Snapshot to wave equation#

Example: Snapshot to full wave equation

A snapshot at \(t = 0\) shows a 5.0 Hz wave traveling left. Find (a) wave speed, (b) phase constant, (c) displacement equation.

[FIGURE: Snapshot graph of sinusoidal wave with labeled axes; indicate \(y_m\), \(\lambda\), and a specific point such as a zero-crossing or peak]

Solution: (a) From \(\lambda\) (read from graph) and \(f = 5\) Hz: \(v = \lambda f\). (b) Use the displacement and slope at \(x = 0\), \(t = 0\) to determine \(\phi\). (c) For leftward travel use \(y = y_m \sin(kx + \omega t + \phi)\) with \(k = 2\pi/\lambda\) and \(\omega = 2\pi f\).

Summary#

Transverse: displacement \(\perp\) propagation; longitudinal: displacement \(\parallel\) propagation.
\(y(x,t) = y_m \sin(kx - \omega t + \phi)\); \(k = 2\pi/\lambda\), \(\omega = 2\pi f\).
Wave speed \(v = \omega/k = \lambda f\); \(kx - \omega t\) \(\Rightarrow\) \(+x\); \(kx + \omega t\) \(\Rightarrow\) \(-x\).
Transverse velocity \(u = \partial y/\partial t\); maximum at \(y=0\), zero at \(y = \pm y_m\).