Download Wave Motion-Classical Physics-Handouts and more Lecture notes Classical Physics in PDF only on Docsity!
r 1
r 2
r 1
Summary of Lecture 20 – WAVE MOTION
- Wave motion is any kind of self-repeating (periodic, or oscillatory) motion that transports energy from one point to another. Waves are of two basic kinds: (a) Longitudinal Waves : the oscill ation is parallel to the direction of wave travel. Examples: sound, spring, "P-type" earthquake waves. (b) Transverse Waves : the oscillation is perpendicular to the direction of wave trave l. Examples: radio or light waves, string, "S-type" earthquake waves.
- Waves transport energy, not matter. Taking the vibration of a string as an example, each segment of the string stays in the same place, but the work done on the string at one end is transmitted to the other end. Work is done in lifting the mass at the other end below.
0
- The height of a wave is called the amplitude. The average power (or intensity) in a wave is proportional to the square of the amplitude. So if ( ) sin( ) is a wave of some kind, then
a t = a ω t − kx
2 0 is the amplitude and^0.
- A sound source placed at the origin will radiate sound waves in all directions equally. These are called spherical waves. For spherical waves the amplitude 1 and
a I a
r
∝ so the power 12. We can easily see why this is so. Consider a source of sound and draw two spheres:
r
1 1 1 1 2 12 1
Let be the total radiated power and I the intensity at , etc. All the power (and energy) that crosses also crosses since none is lost in between the two. We have that, 4
P r r r π r I = P 2 1 22 (^1 2 2 2 1 2 2 ) 2 1
and 4 r I P. But P P P , and so^ I^ r or I^1. I r r
π = = = = ∝
- We have encountered waves of the kind , 0 sin in the previous lecture.
Obviously 0,0 0. But what if the wave is not zero at 0, 0? Then it could be represented by , (^) m sin
y x t y kx t y x t y x t y kx
= − ω = = =
, where is called the phase and is called the phase constant. Note that you can rewrite , either as,
a) , sin ,
or as, b)
m
t kx t y x t
y x t y k x t k
ω φ ω φ φ φ (^) ω
= ⎡^ ⎛^ − ⎞− ⎤
⎢ ⎜⎝^ ⎟⎠ ⎥
( ,^ ) sin^.
The two different ways of writing the same expression can be interpreted differently. In (a) has effectively been shifted to whereas in (b) has been
y x t y m kx t
x x t k
ω φ ω
φ
= ⎡^ − ⎛^ + ⎞⎤
⎢ ⎜⎝^ ⎟⎠⎥
− shifted to. So the phase constant only moves the wave forward or backward in space or time.
- When two sources are present the total amplitude at any point is the sum of the two separate a
t^ φ ω
1 2 2
mplitudes, , , ,. Now you remember that the power is proportional to the of the amplitude, so ( ). This is why happens. In the following we shall see why. J
y x t y x t y x t square P y y interference
1 (^ )^ (^1 )^2 (^ )^ (^2 )
ust to make things easier, suppose the two waves have equal amplitude. So lets take the two waves to be : , sin and , sin The total amplitudes is:
y x t = y m kx − ω t − φ y x t = ym kx − ω t −φ
1 2 1 2
sin sin Now use the trigonometric formula, sin sin 2sin 1 cos to get, 2
m
y x t y x t y x t y kx t kx t B C B C B C
ω φ ω φ
= ⎡⎣^ − − + − − ⎤⎦
+ = + × −
1 2
2 1
, sin sin
2 cos sin. 2
Here is the difference of phases, and
m
m
y x t y kx t kx t
y kx t
ω φ ω φ φ ω φ
φ φ φ φ φ
= ⎡^ ⎛^ Δ ⎞⎤^ × − − ′
⎢⎣ ⎜⎝^ ⎟⎠⎥⎦
Δ = − ′= (^1 2 )
2 1 2
is the sum. So what do 2 we learn from this? That if , then the two waves are in phase and the resultant amplitude is maximum (because cos 0 1). And that if , then the two waves a
φ
φ φ φ φ π
= = + re out of phase and the resultant amplitude is minimum (because cos / 2 0). The two waves have interfered with each other and have increased/decreased their amplitude in these two extreme cas
π =
es. In general cos will be some number that lies between 2 -1 and +1.
⎛ Δ^ φ⎞ ⎜⎝ ⎟⎠
