7.
(a) We write the expression for the displacement in the form y(x, t) = y
m
sin(kx
− ωt). A negative sign
is used before the ωt term in the argument of the sine function because the wave is traveling in the
positive x direction. The angular wave number k is k = 2π/λ = 2π/(0.10 m) = 62.8 m
−1
and the
angular frequency is ω = 2πf = 2π(400 Hz) = 2510 rad/s. Here λ is the wavelength and f is the
frequency. The amplitude is y
m
= 2.0 cm. Thus
y(x, t) = (2.0 cm) sin
62.8 m
−1
x
−
2510 s
−1
t
.
(b) The (transverse) speed of a point on the cord is given by taking the derivative of y:
u(x, t) =
∂y
∂t
=
−ωy
m
cos(kx
− ωt)
which leads to a maximum speed of u
m
= ωy
m
= (2510 rad/s)(0.020 m) = 50 m/s.
(c) The speed of the wave is
v =
λ
T
=
ω
k
=
2510 rad/s
62.8 m
−1
= 40 m/s .