standing wave on a string?

Question

Adjacent antinodes of a standing wave on a string are a distance 15.00 cm apart. A particle at an antinode oscillates in simple harmonic motion with amplitude .8500 cm and period .0750 s. The string lies along the + x - axis and is fixed at x=0.

At time t=0, all points on the string are at their minimum displacement.

Find the displacement of a point on the string as a function of position and time. (in meters) in the form {y(x,t)=}

Find the speed of propagation of a transverse wave in the string. (m/s)

Find the amplitude at a point a distance 3.0 cm to the right of an antinode. (in meters)

thanx in advanced!

its in the form y(x,t)=Acos(kx-wt)

i can't figuer out what to plug where or how to do it....

Answer 1

A = 0.0085 m
Spatial wavelength lambda = 2 * 0.15 m = 0.3 m (because antinodes are 0.5 wavelength apart)
temporal frequency omega1 = 2*pi/0.075 rad/s
spatial frequency omega2 = 2*pi/0.3 rad/m.
y(x,t) = A * sin(omega*t) * sin(omega2*x)
(x measured in m)

To find amplitude at a point 0.03 m in the +x direction from an antinode, solve the spatial location part of the above equation, y(x) = A sin(omega2*x), for x = lambda/4 + 0.3.

Speed of propagation = 0.3 m / 0.075 s = 4 m/s

Answer 2

For a standing wave, the general form is the product of 3 terms:

y(x,t) =
A*(sinusoidal function of x)*(sinusoidal function of time)

The term "A" is the amplitude at an antinode.
Each sinusoid term may be a sine or cosine function, or some combination of the two.

Since the string displacement is zero at x=0, for all times, the x-dependence must be a sine function.

Since the displacement is zero at t=0, (for all x) the time dependence is also sine function.

So we have

y=A * sin(x *2*pi / wavelength) * sin(t * 2*pi / period)

You are given information with which to figure out A, the wavelength, and the period. Putting in those numbers will give you the complete equation. Be aware that adjacent antinodes are 1/2 of one wavelength apart.

To get the speed, use the formula for wave speed in terms of wavelength and frequency (frequency can be figured out from the period).

Finally, the amplitude at any point is:

A*sin(x * 2*pi/wavelength),

Plug in x = 3 cm to find it.

Good luck.

Answer 3

y = A (sin wt + phi) i think is what your looking for