Wave on a string
Any way to calculate the wave speed? What is it likely to depend on?
Amplitude of the wave? Wave length? Mechanical properties of the string? All of those options are plausible, but it turns out the wave speed only depends on mass of the string (rope) and its tension.
Wave on a string
Waves on a string resemble very much harmonic oscillations of a mass on a spring. Tension provides the restoring force, which wants to make the oscillations more frequent. Mass of the string provides the inertia, which slows down oscillations and wave propagation.
Speed of a wave on a string
• Speed of a wave on a stretched string under tension, F
µ = ∆ m/∆ L is the mass per unit length of the string
The wave speed is found as
v=
F
µ
The frequency of oscillations is v 1
f = =
λ λ
F
µ
1
cf. pendulum f =
2π
k
m
F = mload g ≈ 20 N
µ=
How does v change if we increase tension by a factor of 2? make the string twice thicker?
mstring
L
=
mstring
6m
Frequency of a wave on a guitar string
v 1
f = =
λ λ
F
µ
Wavelength, λ = 2L, where L is the length of the string.
If the string has a mass m, µ = m/L, and we have 1
F
1
f =
=
2L m/ L 2
F
F
=
Lm
4 Lm
Piano: thick, heavy, and long strings – low pitch.
Tight strings – high pitch.
Compare: a mass on a spring.
Spring constant
(restoring force)
F
4L
k
m
f =
vs.
k
A propagating wave communicates motion and carries away energy.
We can define wave power as amount of energy carried away from the source (and transported through the medium) per unit time
Wave power: how much energy is transported per unit time.
Measured in Joules/sec, J/s or Watts W. For a harmonic wave propagating along a string
P = FωkA sin (kx − ωt )
2
2
If we average it over time, at any position, x, we find that
< sin (kx − ωt ) > = 1 / 2
2
1
2
Therefore the power averaged over time P = FωkA
2
2
We can plug in k = ω / v and to get
F = µv
1
2 2
P = µ ωA v
2
Let’s take a closer look at the power equation. What kind of sense does it make?
1
1 2
2 2
P = µ ω A v = µv0 v
2
2
µ
is the mass per unit length;
ω
is the angular frequency of oscillations in the wave
A is the amplitude of the oscillations
v is the wave speed
ω A = v0 is the amplitude of the velocity of oscillations of the material of the string along the y­axis
2
0 is the kinetic energy of oscillations per unit length
1
µv
2
The power is proportional to the mass per unit length, to the square of the amplitude of oscillations of the velocity of string material and to the wave speed.
Wave front is a continuous line or a surface connecting nearby wave crests. Plane waves
Wave fronts are flat surfaces for well directed light, radar or sound beams, which propagate in one direction without spreading
Wave fronts are straight lines for ripples on water surface at shore line.
Wave front is a continuous line or a surface connecting nearby wave crests. Wave fronts are spherical surfaces for spherical waves originating from a point source and propagating in 3D space.
Wave fronts are circles for waves on water surface originating from a point source.
Spherical and circular waves becomes less intense as they travel further away from their source, because the same power emitted by the source is spread over larger area (circumference). Wave intensity is the wave power per unit area
P
I=
A
Measured in J
W
or 2
2
s⋅m
m
For a plane wave the intensity remains constant.
For a spherical wave it decreases with the distance, r, from the source, like
P
I=
2
4πr
Power P of the source is 1000 watts.
The energy spreads outward through the air. At a distance R from the source the area A of the spherical front is 4π R2 Intensity = number of joules per second per square meter
= power/area = P/A
Example: 1000 Joules per second of sound energy is generated. Spherical wave fronts spread outward from sound source at the center.
What is the intensity of sound at a distance of 10 m?
R = 10 m
P = 1000 W
I = P / A = P /(4πR 2 ) = 0.8 W/m 2
Is this a rock concert or a whisper?
At the Earth surface the radius of curvature of the wave front is 150,000,000 km,
The intensity is the same everywhere, and the wave is practically flat.
It is a spherical wave on the scale of the Solar system, though.
P
I=
2
4πr
A light bulb is a source of a spherical wave.
If you are 5 ft away from the bulb and walk another 5 ft away the light intensity decreases by a factor of 22 = 4. In terms of intensity of sunlight, it is about the same as going from Venus to mars