Vibrations of Stretched Strings — Formula Sheet

v = √(T/m)

n = v/(2l) = (1/(2l))√(T/m)

m = πr²ρ

T = Mg

n ∝ 1/l or nl = constant

n ∝ √T

n ∝ 1/√m

y1 = a sin[(2π/λ)(vt − x)]

y2 = a sin[(2π/λ)(vt + x) + π] = −a sin[(2π/λ)(vt + x)]

y = −2a cos(2πvt/λ) sin(2πx/λ)

λ = 2l/p

n = (p/(2l))√(T/m)

n = (1/(2l))√(T/m)

n = (1/(2l))√(Mg/(πr²ρ))

Nodes are at x = 0, l/p, 2l/p, 3l/p, ..., l.

Antinodes are at x = l/(2p), 3l/(2p), 5l/(2p), ..., (2p - 1)l/(2p).

The standing-wave equation is y = -2a cos(2pi vt/lambda) sin(2pi x/lambda).

For a given string under a given tension, frequency varies inversely as vibrating length: n ∝ 1/l, or nl = constant.

For a uniform string of given length and material, frequency varies directly as the square root of tension: n ∝ √T.