Assignment 1 solutions

Exercise 3 from assignment 1a.  The sound wave can be described as a pressure wave in space and time as  
Δp = pℴ sin[2π/λ(x-vt)].  In this instance we know the frequency f = 880 Hz, and the velocity v = 345 m/s.  Using 
v = λf we have λ = 0.392 m

v= 345. m/s;
f=880 cycle/s;
λ=v/f

The angular frequency ω = 2πf, the propagation constant k = 2π/λ, and the period T = 1/f

f = 880; 
ω=2Pi f

N[1760 π,3]

λ=0.392 ;
k=2Pi/λ

Precision[16.0285339468867`]

N[MachinePrecision,3]

f=880.;
T=1/f

Thus, we have ω = 5529.2030703180360996942`3. rad/s, k = 16.0 rad/m, T = 1.14 ms (milliseconds)

Exercise 1 from assignment 1b.  We have  v = 345 m/s and λ = 2m.

v=345.;
λ=2.;
f=v/λ
T=1/f
k=2 Pi/λ
ω=2 Pi f

So f  =  172 Hz, T = 5.80 ms, k = 3.14 /m, and ω = 1080 rad/s.

Exercise 2 from assigment 1b.  v = λf,  k = 2π/λ, and ω = 2πf. So λ = 2π/k and f = ω/2π.  Therefore,
v = (2π/k)(ω/2π) = ω/k

Exercise 3 from assignment 1b.  I  plot the two functions and their sum or superposition below.  For purposes of making the plot,  I let α = ωt and let α vary from 0 to 4π radians.

Plot[{Sin[α],Sin[α+Pi/4], Sin[α]+Sin[α+Pi/4]},{α,0.,4Pi}]