Classical Physics I
PHY131
Lecture 27
Waves
Lecture 27
1
Mechanical Waves
A mechanical wave is a “disturbance” that travels through
a medium: particles of the medium undergo displacements
from and/or oscillations about their equilibrium positions
as the “wave” passes by…
– Mechanical waves are possible only because of the restoring forces
acting on the particles that make up the medium…
– Examples:
• transverse pulse in a stretched rope
• longitudinal pulse in a slinky
• water waves are a combination of both…
– Although the participants in the wave motion move around their
equilibrium position, and thus stay “in the same place on average”,
the wave travels on and transports energy and momentum from
one place to the next!
– In a uniform medium, the “disturbance” travels with a
characteristic speed v, which is determined by the medium
properties (density, mass/length, tension, elastic modulus,
temperature, …)
Lecture 27
2
Periodic Waves
a periodic wave is a disturbance (local pressure, transverse
position, etc.) that re-occurs with a certain temporal
periodicity T (= repetition time) and travels as a disturbance
through the medium with a spatial periodicity called “wave
length” λ (= repetition length):
– at any given location on the path of the wave the local medium moves
with period of motion Period T
– The pattern of the disturbance repeats itself along the direction of
its motion with a repetition length called wavelength: λ
– clearly these relate to the propagation speed as:
v = λ/T = λf
T
motion picture (“movie”) of the point x1
t
T
t1
λ
“snapshot” of the wave at time = t1
v
x
λ
Lecture 27
x1
3
Traveling Wave Function y(x,t)
• SHM of “waving” particles of the medium is particularly simple, and
both the movie of the motion of any point, as well as the shape of
the traveling disturbance will be sinusoidal functions.
• Denoting the “disturbance” (e.g. the transverse displacement of a
string) as y, a function of two variables: position x and time t, we
need to find y(x,t):
“movie” of motion
– at a fixed point x = x1 we have:
of point x1
y(x=x1,t) = A cos(ωt+φ)
where A is the maximum displacement, i.e. amplitude, and ω ≡ 2π/T
– at a fixed time t = t1 we have:
“snapshot” of the wave at t1
y(x,t=t1) = A cos(kx+φ’)
– where A is again the amplitude, and k ≡ 2π/λ
do not confuse with
spring constant!!
Lecture 27
4
Traveling Wave Function y(x,t)
Consider a shape y(x,t) that propagates in the x-direction with speed v.
The shape remains the same as it propagates…
– at a point Δx further along in direction of travel, the disturbance will occur
a bit later, delayed by Δt = Δx/v
– Thus, the shape function is y(x,t=0) at time t=0, and travels with velocity v .
At some time later, time t=Δt, the shape remains the same but all values of x
have moved by Δx = vt: y(x,0) = y(x – Δx, Δt) = y(x – vt, Δt)
Thus, the wave function y(x,t) must be simply the function y(x – vt)
For the case of sinusoidal waves we have
y(x – vt) = A cos{k(x – vt)} = A cos(kx – kvt)
– where A is the amplitude,
– where k = 2π/λ and thus kv = kλ/T = 2π/T = ω: y = A cos(kx – ωt)
Thus, displacement at x and t is: y(x,t) = A cos(kx – ωt) = A cos(2πx/λ – 2πt/T)
– where A (and a possible phase-angle φ) are determined by initial conditions,
– and where v= λ/T=ω/k is determined by the properties of the medium…
Note: a wave traveling in the –x direction would be represented by:
y(x,t) = A cos(kx + ωt)
Lecture 27
5
Wave on a String
We are now trying to find the propagation speed of waves, in particular
waves (of small amplitude) traveling on a string under tension FT
– Relevant physical characteristics of the medium (string) are: Tension FT and
mass-per unit-length μ=M/L of the string…
– Dimensional analysis:
• v in m/s, FT in N = kgm/s2, and μ has dimensions kg/m Î
• the only combination of FT, μ giving m/s is √(FT/μ)
Fy(x+dx)
– Full analysis: Consider the net force on a piece
dm = μdx of the string:
dFNet = Fy ( x + dx) − Fy ( x) = FT
⇒
dy ( x + dx)
dy ( x)
− FT
dx
dx
FT
y
FT
2
d y
dt 2
dy ( x + dx) dy ( x)
−
2
d y
d2y
dx
dx
= FT 2
μ 2 = FT
dt
dx
dx
= ( dm ) a y = ( μ dx )
F(x+dx)
ay
Fy(x)
F(x)
x
0
– Note: ax= 0 thus Fx
– General solution of
y ( x, t ) = A f (kx ± ωt + ϕ )
this Wave Equation:
Net=
Lecture 27
shape of the
string y(x,t)
x+dx
with v =
λ
T
=
ω
k
FT
=
8
μ
Superposition Principle
• Note that a cosine/sine function that has ω/k = v =
√(FT/μ) is also good solution…
• In fact, any SUM of cosine and sine functions (lest
they all have the same v) would be a good solution!!
– Fourier Theorem: any repetitive function can be represented
as an (infinite) sum of sine and cosine functions!
• This is a formulation of the “SUPERPOSITION
PRINCIPLE” and follows because the wave equation is
first-power in y(x,t):
– See:
d 2 y1 d 2 y2
d2
y (x, t) + y2 (x, t) ) = 2 + 2
2 ( 1
dt
dt
dt
FT ⎛ d 2 y1 d 2 y2 ⎞ FT d 2
y (x, t) + y2 (x, t) )
=
=
⎜ 2 +
2 ⎟
2 ( 1
dx ⎠ μ dx
μ ⎝ dx
Lecture 27
9