Longitudinal Waves In a longitudinal wave the particle displacement is parallel to the direction of wave propagation. The animation above shows a one­dimensional longitudinal plane wave propagating down a tube. The particles do not move down the tube with the wave; they simply oscillate back and forth about their individual equilibrium positions. Pick a single particle and watch its motion!
The wave is seen as a motion of compressed regions (i.e. it is a pressure wave), which move from left to right.
λ
v = = λf
T
Transverse vs. longitudinal wave
Both propagate from left to right, but cause disturbances in different directions, ∆ y and ∆ x.
∆y (t ) = Ay sin(ωt )
wavelength, λ
Ay
amplitude
∆x(t ) = Ax sin(ωt )
wavelength, λ
Ax
amplitude
Normally the amplitudes of (harmonic) motion of the particles are much smaller than the wavelength.
Longitudinal spring waves
Waves on a Spring
A spring can support both longitudinal and transverse waves.
A wave does not have to be either purely longitudinal or purely transverse. It could be a linear combination of the two.
Harmonic waves are not the only possible type of waves!
A wave can also have a shape of a propagating pulse. True for both transverse and longitudinal waves.
A harmonic wave and a pulse are extreme cases.
The intermediate case is a wave train – a finite duration sinusoidal. How do we describe a harmonic wave mathematically?
y
x
λ
• Features to incorporate:
in any point in space the wave produces harmonic oscillations of a type:
y (t ) = Ay cos(ω t + ϕ ) ω
­ angular frequency
ϕ
­ phase
if we “freeze” the wave, we will see a harmonic function in space
y ( x) = Ay cos(kx + ϕ )
k
what is this ? if we freeze the wave and move 1 wavelength along it, we are supposed λ
to see the same level of disturbance y
kλ = 2π
y ( x + λ ) = Ay cos(k ( x + λ ) + ϕ ) = Ay cos(kx + 2π + ϕ ) = y ( x)
Therefore, it must be so that y
x
λ
y (t ) = Ay cos(ω t + ϕ )
y ( x) = Ay cos(kx + ϕ )
ϕ ­ phase kλ = 2π ⇒ k = 2π / λ
ω
­ angular frequency
k
the wave number measured in m­1. What is the meaning of it?
If we freeze the wave and ride along it, we periodically will bump into crests.
k / 2π
tells us how many times per meter it is going to happen ω / 2π = f
k
k
tells us how many times per second we are going to fill a crest if we do not move but rather bob on the wave
ω
is pretty much the same for space as is for time!
tells us the phase change per meter and is sometimes called the spatial frequency
y
x
y (t ) = Ay cos(ω t + ϕ )
ω = 2π / T
T is period in time
λ
y ( x) = Ay cos(kx + ϕ )
k = 2π / λ
λ is period in space
How do we unite the two equations (in time and in space)?
y ( x, t ) = Ay cos(ω t − kx)
x0
ϕ = − kx0
Considering only one point in space, , means taking y ( x0 , t ) = Ay cos(ω t + ϕ )
t0
Freezing it in time, , means taking ϕ = −ω t 0
y ( x, t 0 ) = Ay cos(kx + ϕ )
y
x
λ
y ( x, t ) = Ay cos(ω t − kx)
­ equation of a harmonic wave
ω = 2π / T = 2π f
k = 2π / λ
λ = v/ f
k = 2π f / v = ω / v
ω = kv
y ( x, t ) = Ay cos(kx − ω t ) = Ay cos(kx − kvt ) =
= Ay cos[k ( x − vt )]
A crest corresponds to a point, where k ( x −
Therefore position of the crest is given by vt ) = 0
x = vt
y ( x, t ) = 0.1cos (3t − 2 x)
­ equation of a harmonic wave on a string. All in SI units.
What is the maximal velocity of an element of the string?
k=2
ω=3
the wave speed
v = ω / k = 2/3
m/s
Velocity of an element of the string:
dy ( x, t )
v ( x, t ) =
= − 0.3sin (3t − 2 x)
dt
Maximal velocity of an element of the string: 0.3 m/s
y ( x, t ) = Ay cos(kx − ω t )
­ equation of a harmonic wave
y ( x, t ) = Ay cos[k ( x − vt )]
­ the same equation rewritten in a form emphasizing propagation and wave speed
y ( x, t ) = Ay cos[k ( x + vt )]
­ what would this one stand for?
v
−v
is changed to , which means that the wave is propagating in the negative x­direction, from right to left
In this case, the location of a crest is given by
x + vt = 0
⇒
x = − vt
cos[k ( x + vt )] = 1
How can we describe a pulse? (not a harmonic wave)
Generic equation for a wave traveling in positive x­direction with wave speed v:
y ( x, t ) = f ( x − vt )
Here can be ANY function. The type of the function f (x )
f (x )
specifies the shape of the wave. How do we know it is a propagating (traveling) wave?
y
x
(the disturbance) depends on and in a VERY SPECIAL WAY: t
it only depends on x − vt
x − vt
y
Therefore the disturbance is the same as long as is constant, say x − vt = x0
x − vt = x0
⇒
x = vt + x0
y ( x, t ) = y ( x0 ,0)
A point of constant disturbance, y(x0) , (crest, trough, etc.) moves at the wave speed, v Example: a bell­shaped (Gaussian) curve with a peak at x = 0
y ( x) = f ( x ) = exp(− x 2 )
A bell­shaped (Gaussian) curve with a peak at x = a
y ( x) = exp[−( x − a ) ]
2
What if the peak is moving along the x­
axis with a speed ?
v
We can plug in and get
a = vt
y ( x, t ) = f ( x − vt ) = exp[−( x − vt ) 2 ]
The answer we arrived at:
y ( x, t ) = f ( x − vt ) = exp[−( x − vt ) ]
2
How do we understand it?
x = vt0
y(x,t) is the value of the disturbance at the point and time of interest, x, and t
How does the profile of the disturbance y(x,t) look at time t0?
y ( x, t 0 ) = exp[−( x − vt0 ) ]
a Gaussian function with maximum at x = vt 0
It is 2
As usual for a wave, the position of the maximum is given by xmax = vt
The position of the maximum, the crest, moves at the wave speed, v 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 both the oscillations and wave propagation.