Question

Standing waves on a string are generated by oscillations having amplitude 0.005 m, angular frequency 942 rad/s, and wave number 0.750p rad/m.

a.) What is the equation of the standing wave?
b.) At what distances from x=0 are the nodes and antinodes?
c.) What is the frequency of a point on the string at an antinode?
d.) If the string is 4m long, how many nodes are there?

Answer 1

y(x,t) =|2•A•coskx|•cosωt,
y(x,t) =|2•0.005•cos(0.750•π•x)| •cos(942•t),
Antinods:
x=m• (λ/2), m=0,1,2,...
The 1st antinode at the origin (m=0),
the 2nd antinode at λ/2 (m=1).
Distance from origin = λ/2
Nodes:
x= (m+½)• (λ/2).
The 1st node at λ/4 (m=0),
the 2nd antinode at 3λ/4 (m=1)
Distance from origin = λ
The frequency is f=ω/2π=942/2 π =149.9 Hz.
λ=2π/k=2π/0.75 π=8/3 (meters).
If the antinode is at the origin, then we have
3 nodes on the distance 4 m at the points:
x=λ/4, 3λ/4, 5λ/4,
and 4 antinodes at
x= 0, λ/2, λ, 3λ/2.