Let L = ½∑_{ij}[T_{ij}(dq_{i}/dt)(dq_{j}/dt) - k_{ij}q_{i}q_{j}]
with T_{ij} = T_{ji}, k_{ij} = k_{ji}.

Then solutions of the form q_{j} = Re(A_{j}e^{iωt}) can be found.

We can find the
ω^{2}
from det(k_{ij} - ω^{2}T_{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_{α}^{2}T_{ij}]A_{jα}
= 0

to find the A_{jα}.

[While the secular equation det(k_{ij}-ω^{2}T_{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_{α}A_{jα}exp(iω_{α}t)).

## Simple Harmonic Motion: |
x(t) = Acos(ωt + Φ),
v(t) = dx(t)/dt = -ωAsin(ωt + Φ), a(t) = d ^{2}x(t)/dt^{2} = -ω^{2}Acos(ωt + Φ)
= -ω^{2}x. |

Energy: | K = ½mv^{2}, U = ½kx^{2}, E = K + U = ½kA^{2}. |

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_{max}cos(ωt +
Φ),
ω^{2 }= g/L (small oscillations). |

## Mechanical waves: |
wave equation: d^{2}y(x,t)/dx^{2} = (1/v^{2})d^{2}y(x,t)/dt^{2}. |

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) = nf_{1}.Tube with one closed end: f _{n }= nv/(4L) = nf_{1}, n =
odd. |

Doppler effect: | f = f_{0}(v-v_{o})/(v-v_{s}), f = observed frequency, f _{0} = 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. |