A traverse periodic wave described by the expression y=5sin (pi x/3 - 30pi t) (where y and x are in meters and t is in seconds) is established on a string.
By reflecting this wave from a rigid support, standing transverse waves are set up on a stretched string. It is observed that the string is vibrating in its THIRD HARMONIC. What is the length in meters of the string?
Question 5 options:
A)
3
B)
6
C)
9
D)
12
E)
15
To find the length of the string, we need to consider the relationship between the length of a string and the standing waves produced on it.
In this case, we are observing standing transverse waves vibrating in the third harmonic. The third harmonic is also known as the third overtone, where the fundamental frequency is tripled.
The fundamental frequency of a standing wave on a string is given by:
f1 = v / 2L
Where f1 is the fundamental frequency, v is the wave speed, and L is the length of the string.
Since we are observing the third harmonic, the frequency will be three times the fundamental frequency:
f3 = 3f1
To find the length of the string, we need to solve for L in terms of f3:
f3 = v / 2L
Rearranging the equation, we get:
L = v / (2f3)
Now, we need to determine the wave speed, which is the velocity of the wave.
The wave speed is given by:
v = λf
Where λ is the wavelength and f is the frequency.
From the given expression for y = 5sin (πx/3 - 30πt), we can see that the wave travels one wavelength in the distance covered by x when t = 1.
So, the wavelength is given by:
λ = x
Now, we can substitute the values into the equation for L:
L = v / (2f3)
L = (λf) / (2f3)
L = (xf) / (2f3)
L = x / (2/3)
L = 3x/ 2
From the options provided, we need to find a value of L that satisfies the equation.
Looking at the options:
A) 3
B) 6
C) 9
D) 12
E) 15
We can see that option C) 9 satisfies the equation L = 3x/2.
Therefore, the length of the string is 9 meters.