Wave Motion (II)

Text sections 16.2 – 16.3, 16.6

•Sinusoidal waves

Practice: Chapter 16, Objective Questions 1, 3, 8 Conceptual Question 5 Problems 9, 11, 13, 27 Non-dispersive waves:

y (x,t) = f (x ±vt)

+ sign: wave travels towards –x - sign: wave travels towards +x f is any (smooth) function of one variable.

eg. f(x) = A sin ( kx) is is y y x = kv .,

] ω eg are constants) are

k and with k(x –k(x vt) A ( ) t sin[

ω (kx) kx – kx sin direction, the displacement displacement the direction, sin ( sin +x y (x,t) -(x = f y (x,t) vt) y (x,t) = f(x –y (x,t) A = vt) A = y (x,t) f(x) = y(x,0) = A A = y(x,0) = f(x) Then or y A -A For sinusoidal waves, the shape shape a is sine function, the waves, sinusoidal For SineWaves

For a wave in travelling wave the a For given by The displacement of a particle at a fixed location x is a sinusoidal function of time – i.e., simple harmonic motion:

y = A sin ( kx – ωt) = A sin [ constant – ω t ]

The “angular frequency” of the particle motion is ω ; the initial phase is kx (differentfor different particles).

Review: Recall from last term, SHM is described by functions of the form y(t) = A cos (ω t+ φ) = A sin (π/2 –φ –ω t) , etc., with ω = 2 πf

“angular frequency” frequency: radians/sec cycles/sec (=hertz) Sine wave: y (x,t) = A sin( kx – ω t )

y λ A v x -A

λ (“lambda”) is the wavelength (length of one complete wave); and so (kx) must increase by 2π radians (one complete cycle) when x increases by λ. So kλ = 2 π, or

k = 2 π / λ The most general form of sine wave is y = Asin( kx ± ωt + φ)

amplitude “phase” y(x,t) = A sin (kx ±ω t + φ )

phase constant φ angular wavenumber k = 2 π / λ angular frequency ω = 2 πf

The wave speed is v = 1 wavelength / 1 period, so v = f λ = ω/k Particle Velocities

particle displacement, y (x,t )

particle velocity, vy = dy/dt (x held constant)

(Note that vy is not the wave speed v ! ) 2 dv y d y Acceleration, a = = y dt dt 2

∂ 2 y (actually, these are partial derivatives: , etc.) ∂t 2 “Standard” sine wave: y = A sin( kx ± ωt + ϕ ) dy v = = ±ωA cos( kx ± ωt + ϕ ) y dt dv a = y = −ω 2 Asin( kx ± ωt + ϕ ) y dt = −ω 2 y

maximum displacement, ymax = A maximum velocity, vmax = ω A 2 maximum acceleration, amax = ω A y Quiz A a b e x d -A c

Shown is a picture of a wave, y=A sin (kx − ω t) , at time t=0 .

i) Which particle moves according to y=A cos (ω t ) ?

ii) Which particle moves according to y=A sin (ω t) ?

iii) If ye(t)=A cos (ω t+ φe ) for particle e, what is φe ? Wave Velocity The wave velocity is determined by the properties of the medium; for example,

1) Transverse waves on a string: tension F v = = T wave mass/unit length µ (proof from Newton’s second law)

2) Electromagnetic wave (light, radio, etc.) 1 v = = c µoε o (proof from Maxwell’s Equations for E-M fields) Exercise

What are ω and k for a 99.7 MHz FM radio wave? Wave Velocity FT dm

dx FT

Exercise: show that the net vertical force on the short element of rope (of length dx) is, approximately, given by ∂2 y dF = F dx net T ∂x2 T F y 2 t 2 ∂ ∂ ) dx µ ( = a ) dm

dx ( dm = dx 2 y 2 x ∂ ∂ T F = is the linear mass density. mass linear theis T net

F µ dF where Newton’s second law is second law Newton’s Wave equation Rearranging, we get the wave equation: ∂ 2 y  F  ∂2 y =  T  ∂t 2  µ  ∂x2

Exercise: apply this to our standard sine-wave expression for y(x,t), to show that

F ω 2 T = = v2 µ k 2 wave y Quiz A

x a -A c b

Which particle has the largest particle acceleration at this moment? A) A B) B C)C D) they all have a=0 for a nondispersive wave. Example

y x string: 1 gram/m; 2.5 N tension

vwave Oscillator: 50 Hz, amplitude 5 mm

Find: y (x, t) vy (x, t) and maximum speed ay (x, t) and maximum acceleration