Calculating Properties of a Standing Wave

[sonutulsiani]

Homework Statement

The wave function for a certain standing wave on a string that is fixed at both ends is given by
y(x, t) = (5.00 cm) sin (2.50 m^(− 1)x) cos (500 s^(− 1)t)


A standing wave can be considered as the superposition of two traveling waves.

1.
a. Give the speed in m/s rounded to a whole number.
b. Give the amplitude in cm rounded to one decimal place



2. What is the distance between successive nodes on the string? Give your answer in m rounded to two decimal places.


3. What is the shortest possible length of the string? Give your answer in m rounded to two decimal places.

Homework Equations

The Attempt at a Solution

I got all the answers:

1a. 200m/s
1b. 5.0 cm
2. 1.26 m
3. 1.26 m

But I am sure about none of them, please check if right or not? If it's not, then please help me out. Thanks !

 

[ideasrule]

Seems good!

 

[kerimek]

1a. The speed of a wave on a string is given by the equation v = √(T/μ), where T is the tension in the string and μ is the linear mass density. In this case, we are not given the values for T or μ, so we cannot calculate the exact speed. However, we can assume that T is some typical value for a string, such as 10 N, and μ is the linear mass density of a string, which is typically around 0.01 kg/m. Plugging these values into the equation, we get v = √(10/0.01) = √1000 = 31.62 m/s. Rounded to the nearest whole number, the speed is 32 m/s.

1b. The amplitude of a wave is the maximum displacement from the equilibrium position. In this case, the amplitude is given as 5.00 cm, so the amplitude is 5.0 cm rounded to one decimal place.

2. The distance between successive nodes on a string is equal to half the wavelength of the standing wave. The wavelength is given by λ = 2L/n, where L is the length of the string and n is the number of nodes. In this case, there is only one node, so n = 1. The shortest possible length of the string will occur when n = 2, so we can plug this into the equation to get λ = 2L/2 = L. Therefore, the distance between successive nodes is equal to the length of the string, which is given as 2.50 m. Rounded to two decimal places, the distance between successive nodes is 2.50 m.

3. The shortest possible length of the string will occur when n = 2, as mentioned in the previous question. Plugging this into the equation L = λ/2, we get L = 2.50/2 = 1.25 m. Rounded to two decimal places, the shortest possible length of the string is 1.25 m.