PHY 102: Waves & Quanta

Topic 4

Standing Waves

John Cockburn (j.cockburn@... Room E15)

•Wave reflection at boundaries

•Principle of superposition, interference

•Standing waves on a string

•Normal modes

Reflection of a wave pulse at a boundary

Fixed end Free end

Pulse incident from right is reflected from the boundary at left

HOW the pulse is reflected depends on the boundary conditions

For fixed end, reflected pulse is inverted

For free (in transverse direction) end, reflected pulse is same way up.

time

Frictionless sliding ring

Reflection of a wave pulse at a boundary

Behaviour at interface can be modelled as sum of two pulses moving in opposite directions at the interface:

Transverse displacement always 0 at interface

“fixed end” “free end”

Transverse force

always 0 at interface

Principle of superposition

When 2 (or more) waves overlap in time/space, the total effect is just the algebraic sum of the individual wave functions:

),(..........),(),(),(),( 321 txytxytxytxytxy ntotal

2

2

22

2 ),(1),(

t

txy

vx

txy

(must be so, because wave equation is linear: if y1(x,t) and y2(x,t) are both solutions, for example, then y1+y2 must also be a solution)

Formation of standing wave on a string

Pink line represents wave travelling from right to left along the string.

Blue line represents wave travelling from left to right.(wave reflection at boundaries)

•Black line = sum of left and right-travelling waves = STANDING WAVE

•Constructive interference of waves at ANTINODE of standing wave (max displacement)

•Destructive interference of waves at NODE of standing wave (zero displacement)

•Distance between successive nodes/antinodes = λ/2

Mathematical formulation of standing wave

Wave moving right to left (pink wave)

)cos(),(1 tkxAtxy

Wave moving left to right (blue wave)

)cos(),(2 tkxAtxy

Total wave function (black wave):

Mathematical formulation of standing wave

)sin()sin2(),( tkxAtxy

Mathematical formulation of standing wave

)sin()sin2(),( tkxAtxy

amplitude dependson position

Zero y-displacement (node) when sin(kx) = 0

Maximum y-displacement (y=2A) when sin(kx)=+/- 1……..

Comparison between standing wave and travelling wave

Travelling wave

particles undergo SHM

all particles have same amplitude

all particles have same frequency,

adjacent particles have different phase

Standing wave

particles undergo SHM

adjacent particles have different amplitude

all particles have same frequency

all particles on same side of a node have same phase. Particles on opposite sides of

node are in antiphase

Some very basic physics of stringed instruments……….

The fundamental frequency determines the pitch of the note.

the higher harmonics determine the “colour” or “timbre” of the note.

(ie why different instruments sound different)

Fundamental wavelength = 2L

From v = fλ,

f1= v/2L

So, for a string of fixed length, the pitch is determined by the wave velocity on the string…..

Example Calculation

The string length on standard violin is 325mm. What tension is required to tune a steel “A” string (diameter =0.5mm) to correct pitch (f=440Hz)?

Density of steel = 8g cm

Changing the “harmonic content”

string plucked here

Plucking a string at a certain point produces a triangular waveform, that can be built up from the fundamental plus the higher harmonics in varying proportions.

Plucking the string in a different place (or even in a different way) gives a different waveform and therefore different contributions from higher harmonics (see Fourier analysis)

This makes the sound different, even though pitch is the same…………………