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Unformatted text preview: ion x: 2 ∂y
∂ t2 Acceleration felt e.g. by an ant at position x. Segment of string from x to x+dx has mass: µdx
Linear mass density of the string
Wednesday, October 9, 2013 Derive wave eqn.
Follows from F=ma, applied to string elements.
y (t, x) τ
dFy = τ ( |x+dx −
∂x Tension (“F” in book). ∂2y
dFy = (dm)ay = (µdx) 2
Linear mass density Wednesday, October 9, 2013 dy
dFy = τ 2 dx
∂x Equating gives the 1d wave eqn, with v= τ
µ Ex: verify
work, DA! Wave energy, power
y (t, x) τ
x dy Force exerted on string to
the right, by the string to the left Method 1:
(x, t) · (x, t) = Fy (x, t)vy (x, t) = −τ ∂ y ∂ y
P (x, t) = F
∂x ∂t Method 2: 1
k (x, t) = µ
∂t 2 dE dx
= (k + u)v
dx dt Wednesday, October 9, 2013 1
u(x, t) = τ
∂x 2 energy
densities Both methods give the same
answer (using the wave eqn): Wave power, cont.
P (x, t) = F (x, t) · (x, t) = Fy (x, t)vy (x, t) = −τ
∂x ∂t y = A cos(kx − ω t)
P = τ k ω A2 sin2 (kx − ω t)
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This note was uploaded on 02/11/2014 for the course PHYSICS 2C taught by Professor Hicks during the Fall '09 term at UCSD.
- Fall '09