Question

Two sinusoidal waves traveling in opposite directions interfere to produce a standing wave with the following wave function, where x is in meters and t is in seconds. y = (3.00 m) sin(0.200x) cos(2006) Determine the wavelength of the interfering waves. What is the frequency of the interfering waves? Hz Find the speed of the interfering waves. m/s

Answer 1

compare the given equation with y = A*sin(k*x) cos(w*t)

A = 3.0 m
k = 0.2 rad/m
w = 200 rad/s

a) we know, k = 2*pi/lamda

==> lmada = 2*pi/k

= 2*pi/0.2

= 31.4 m

b) f = w/(2*pi)

= 200/(2*pi)

= 31.83 Hz

c) v = w/k

= 200/0.2

= 1000 m/s

here 2)

y1 = 6.00 sin π(x + 0.600t)
y1 = 6 (sin πx*cos 0.600t + cosπx*sin 0.600t)

y2 = 6.00 sin π(x − 0.600t)
y2 = 6 {sin πx*cos 0.600t - cosπx*sin 0.600t}

y = y1 + y2
y = 12 (sin πx)*(cos(0.600t) ) --------------------(1)

Part A:
x = 0.180 cm

for eqn 1 to be max , cos(0.600t) = 1,
From eqn 1,
ymax = 12*sin( π*0.180)
ymax = 12*0.535
ymax = 6.42 cm

Part B:
x = 0.620 cm

for eqn 1 to be max , cos(0.600t) = 1,
From eqn 1,
ymax = 12*sin( π*0.620)
ymax = 12*0.929
ymax = 11.15 cm

Part C:
x = 1.40 cm

for eqn 1 to be max , cos(0.600t) = -1,
From eqn 1,
ymax = 12*sin( π*1.40)
ymax = -12* -0.951
Ymax = 11.41cm

Part D:
the antidote occur when
x = n*lamda/4 , n = 1,2,3,4......

for combined wave function we find :
k = 2pi/lamda or lamda = 2

the smallest value of x corresponding to antidote are given by n = 1 , 3 , 5 ..

x1 = lamda/4 = 0.5 cm
x2 = 3*lamda/4 = 1.50 cm
x3 = 5*lamda/4 = 2.50 cm