Oscillations and waves

Coupled small oscillations

Let L = ½∑_ij[T_ij(dq_i/dt)(dq_j/dt) - k_ijq_iq_j] with T_ij = T_ji, k_ij = k_ji.

Then solutions of the form q_j = Re(A_je^iωt) can be found.
We can find the ω² from det(k_ij - ω²T_ij) = 0.
For a system with n degrees of freedom, n characteristic frequencies ω_α can be found.
Some frequencies may be degenerate.
For a particular frequency ω_α we solve

∑_j[k_ij - ω_α²T_ij]A_jα = 0

to find the A_jα.

[While the secular equation det(k_ij-ω²T_ij) = 0 can in principle always be solved, it is often simpler to find the normal modes by using physical insight and noting the symmetries of the system.]

The most general solution for each coordinate q_j is a sum of simple harmonic oscillations in all of the frequencies ω_α.

q_j = Re∑_α(C_αA_jαexp(iω_αt)).

Simple oscillations and waves

Simple Harmonic Motion:	x(t) = Acos(ωt + Φ), v(t) = dx(t)/dt = -ωAsin(ωt + Φ), a(t) = d²x(t)/dt² = -ω²Acos(ωt + Φ) = -ω²x.
Energy:	K = ½mv², U = ½kx², E = K + U = ½kA².
A mass on a spring:	ω = (k/m)^½, T = 2π(m/k)^½, f = (1/(2π))(k/m)^½.
A simple pendulum:	θ(t) = θ_maxcos(ωt + Φ), ω²= g/L (small oscillations).
Mechanical waves:	wave equation: d²y(x,t)/dx² = (1/v²)d²y(x,t)/dt².
Sinusoidal waves:	y = Asin(kx ± ωt + Φ), k = 2π/λ, ω = 2π/T = 2πf, v = λ/T = λf.
Waves in a string:	v = (F/μ)^½, F = tension in the string, μ = mass per unit length.
Standing waves:	String and tube with two open ends: f_n= nv/(2L) = nf₁. Tube with one closed end: f_n= nv/(4L) = nf₁, n = odd.
Doppler effect:	f = f₀(v-v_o)/(v-v_s), f = observed frequency, f₀ = frequency of source, v = wave velocity, v_o = velocity of observer, v_s = velocity of source. v_o and v_s are not the speeds, but the components of the observer's and the source's velocity in the direction of the velocity of the sound reaching the observer.