16-7 Standing Waves and Resonance#

Prompts

Two identical waves travel in opposite directions on a string. What is the form of the resultant wave? Does it travel?
What are nodes and antinodes? Where do they occur for the standing wave $y' = [2y_m \sin kx] \cos \omega t$?
A string is fixed at both ends. Why must there be a node at each end? What wavelengths can form standing waves on a string of length $L$?
Write the resonant frequencies $f_n$ for a string of length $L$ with fixed ends. What is the fundamental frequency? The second harmonic?
Distinguish hard reflection (fixed end) from soft reflection (free end). Where is the node or antinode in each case?

Lecture Notes#

Overview#

Standing waves arise when two identical waves travel in opposite directions and interfere. The pattern does not travel—nodes and antinodes stay fixed.
Nodes: positions of zero displacement. Antinodes: positions of maximum amplitude.
On a string fixed at both ends, only certain wavelengths (and frequencies) produce standing waves—resonance. The resonant frequencies are $f_n = n v/(2L)$.

Standing waves from opposite-going waves#

Two waves of the same amplitude and wavelength, traveling in opposite directions:

(105)#\[ y_1 = y_m \sin(kx - \omega t), \qquad y_2 = y_m \sin(kx + \omega t) \]

By superposition: $$ y'(x,t) = y_1 + y_2 = [2y_m \sin kx] \cos \omega t $$ (eq-standing-resultant)

This is not a traveling wave (no $kx \pm \omega t$ form). The pattern is stationary.
Position-dependent amplitude: $|2y_m \sin kx|$ varies with $x$; the $\cos \omega t$ factor gives the time dependence.

Nodes and antinodes#

Nodes (zero displacement, always at rest): $\sin kx = 0$ $\Rightarrow$ $kx = n\pi$ $\Rightarrow$

(106)#\[ x = n\,\frac{\lambda}{2}, \quad n = 0, 1, 2, \ldots \]

Adjacent nodes are separated by $\lambda/2$.

Antinodes (maximum amplitude): $|\sin kx| = 1$ $\Rightarrow$ $kx = (n + \frac{1}{2})\pi$ $\Rightarrow$

(107)#\[ x = \left(n + \frac{1}{2}\right)\frac{\lambda}{2}, \quad n = 0, 1, 2, \ldots \]

Antinodes lie halfway between nodes. At an antinode, each element undergoes SHM with amplitude $2y_m$.

Reflection at boundaries#

Hard reflection (fixed end): The string cannot move at the support. A node must occur there. The reflected pulse is inverted (phase flip) so incident and reflected cancel at the fixed point.

Soft reflection (free end, e.g., ring sliding on a rod): The end can move freely. An antinode occurs there. The reflected pulse has the same sign as the incident—no inversion.

End type	Reflection	Node or antinode at end
Fixed (hard)	Inverted	Node
Free (soft)	Same sign	Antinode

Poll: Pulse reflection at fixed end

A symmetric pulse approaches the right end of a string fixed at both ends. Which best represents the string after the pulse has fully reflected?

[FIGURE: Before — pulse moving right toward wall; After — inverted pulse moving left]

(A) Upright pulse moving left
(B) Inverted pulse moving left
(C) Two pulses (original + reflected)
(D) Flat string

Poll: Two pulses meeting after reflection

Two identical pulses move toward opposite fixed ends. After both reflect and meet in the middle, which shape is possible?

[FIGURE: Two pulses approaching from left and right; then overlapping; then separating after passing]

(A) Single large pulse
(B) Momentarily flat, then two pulses re-emerge
(C) Destructive cancellation forever

Poll: Standing wave speed from pattern

A string fixed at both ends (1 m apart) vibrates at 300 Hz. A strobe photo shows 3 loops (3 antinodes). What is the wave speed?

(A) 50 m/s
(B) 100 m/s
(C) 150 m/s
(D) 200 m/s
(E) 300 m/s

Resonance on a string with fixed ends#

A string of length $L$ fixed at both ends must have a node at each end. Only wavelengths that fit an integer number of half-wavelengths between the ends support standing waves:

(108)#\[ L = n\,\frac{\lambda}{2} \quad \Rightarrow \quad \lambda = \frac{2L}{n}, \quad n = 1, 2, 3, \ldots \]

Resonant frequencies (using $f = v/\lambda$):

(109)#\[ f_n = \frac{v}{\lambda} = n\,\frac{v}{2L} = n f_1 \]

$n = 1$: fundamental mode (first harmonic); $f_1 = v/(2L)$; one loop.
$n = 2$: second harmonic; two loops.
$n = 3$: third harmonic; three loops; etc.

Important

Resonance occurs when the driving frequency matches the natural frequency of a standing-wave mode. For other frequencies, the interference of reflected waves does not produce a sustained standing pattern.

Summary#

Standing wave: $y' = [2y_m \sin kx] \cos \omega t$—two opposite-going waves interfere.
Nodes at $x = n\lambda/2$; antinodes at $x = (n + \frac{1}{2})\lambda/2$.
Fixed end → node, inverted reflection. Free end → antinode, same-sign reflection.
Resonance: $\lambda = 2L/n$, $f_n = n v/(2L)$; $n = 1$ is the fundamental.