The velocity of a wave on a particular string is 21.8 m/s and the string is 3 meters long. What are the 3 lowest harmonic frequencies that with produce a standing wave?
To find the three lowest harmonic frequencies that will produce a standing wave on the string, we need to understand the relationship between the velocity, the length of the string, and the harmonic frequency.
First, let's recall the formula for the velocity of a wave on a string:
velocity = frequency * wavelength
Where:
- velocity is the speed at which the wave travels (given as 21.8 m/s)
- frequency is the number of complete cycles or oscillations of the wave per second (in Hz) - what we want to find
- wavelength is the distance between two corresponding points on the wave (in meters)
In this case, the length of the string is given as 3 meters. For the first harmonic, the length of the string is divided into two equal parts (half of the wavelength). So, the wavelength for the first harmonic (n = 1) is twice the length of the string, λ₁ = 2 * 3 = 6 meters.
Using the formula for velocity:
21.8 m/s = frequency * 6 m
Simplifying the equation, we can solve for the frequency:
frequency = 21.8 m/s / 6 m = 3.63 Hz
This gives us the frequency of the first harmonic, f₁ = 3.63 Hz.
For the second harmonic (n = 2), the length of the string is divided into three equal parts. So, the wavelength for the second harmonic is equal to the length of the string, λ₂ = 3 meters.
Using the formula for velocity:
21.8 m/s = frequency * 3 m
Solving for frequency:
frequency = 21.8 m/s / 3 m = 7.27 Hz
Thus, the frequency of the second harmonic is f₂ = 7.27 Hz.
For the third harmonic (n = 3), the length of the string is divided into four equal parts. So, the wavelength for the third harmonic is three-quarters of the length of the string, λ₃ = (3/4) * 3 = 2.25 meters.
Using the formula for velocity:
21.8 m/s = frequency * 2.25 m
Solving for frequency:
frequency = 21.8 m/s / 2.25 m = 9.69 Hz
Thus, the frequency of the third harmonic is f₃ = 9.69 Hz.
Therefore, the three lowest harmonic frequencies that will produce a standing wave on the string are:
- f₁ = 3.63 Hz
- f₂ = 7.27 Hz
- f₃ = 9.69 Hz