Waves
General Properties
Waves transmit energy
Result from a sequence of particles in SHM
Two general categories
(1) Longitudinal – Particle motion along
direction of wave propagation (p-waves,
compressional waves); Compressions and
rarefactions
(2) Transverse – Particle motion transverse to
direction of wave propagation (s-waves,
shear waves)
Examples:
♦ Earthquake waves – Locating earthquakes;
differences between body and surface waves
♦ Sound waves – longitudinal or transverse?
♦ Light (EM) waves – Polarization demonstrates
they are transverse waves
♦ Water waves
Velocity of waves: v = fλ
But what physical properties of the wavetransporting medium affect the wave velocity?
Answer by considering waves on a string.
Traveling Wave Equation
Force analysis of piece of deformed medium (string)
stretched in x-direction
∂2y µ ∂2y
=
2
τ ∂t 2
∂x
Solution
y ( x , t ) = A sin( kx − ω t )
Interpretations
(1) Stop time and advance x by one wavelength λ.
kx=kλ=2π radians
k=2π/λ rad/m (k called the wavenumber)
(2) Stay in one place, advance time t by one period T.
For example, y(0,t) = Asin(-ωt)
= -Asin(ωt)
Advancing t by T
ωT= 2π radians
ω=2π/T rad/s (called the angular frequency)
(3) ‘Ride’ one spot of wave by keeping the argument
of sin(kx-ωt) constant.
kx-ωt = constant
x = constant + (ω/k)t
Velocity of wave given by:
λ
dx ω 2π λ
v=
= =
= = fλ
dt
k
T 2π T
But ω and k (or f and λ) are not independent of
each other. Set by the physical conditions of
medium, since
ω2
τ
=
2
µ
k
v=
τ ω
=
µ k