An organ pipe open at both ends has a harmonic with a frequency of 480 Hz. The next higher harmonic in the pipe has a frequency of 540 Hz.

Calculate the frequency of the fundamental and the length of the pipe.

I know 540/480 is 9/8. I have no idea where to go from here.

To determine the frequency of the fundamental and the length of the pipe, we can use the relationship between the lengths and frequencies of harmonics in a pipe open at both ends.

In a pipe open at both ends, the harmonic frequencies are given by:

f = (n * v) / (2L)

Where:
f is the frequency of the harmonic
n is the harmonic number (1 for the fundamental, 2 for the first overtone, 3 for the second overtone, etc.)
v is the speed of sound
L is the length of the pipe

We are given that the first overtone (second harmonic) has a frequency of 540 Hz and the fundamental (first harmonic) has a frequency of 480 Hz.

Let's first calculate the speed of sound (v) using the known frequency (f) and harmonic number (n) for the fundamental:

480 = (1 * v) / (2L)

Since the pipe is open at both ends, the fundamental has a node at each end. This means that the length of the pipe (L) is equal to half of a wavelength (λ/2) of the fundamental frequency.

So, the length of the pipe (L) is given by:

L = λ/2

Substituting the speed of sound (v) for the wavelength (λ) in the equation of the fundamental frequency:

480 = (1 * v) / (2 * L)

Now let's calculate the length of the pipe (L). Divide both sides of the equation by 480:

1 / 480 = 1 / (2 * L)

By rearranging the equation, we can isolate L:

L = 1 / (2 * (1 / 480))

L = 1 / (2 / 480)

L = 240 cm

Now, let's calculate the frequency of the fundamental (f1) using the length of the pipe (L) and the speed of sound (v):

f1 = (1 * v) / (2 * L)

Substituting the known value of L:

f1 = (1 * v) / (2 * 240)

To find the value of v, we need more information. The speed of sound depends on factors such as temperature and humidity. Without that information, it is not possible to calculate the exact value for v.

However, based on the ratios of the given frequencies, we can calculate the ratio of the speeds of sound:

(540 / 480) = [(1 * v) / (2 * 240)] / [(1 * v) / (2 * L)]

(540 / 480) = L / 240

(540 / 480) = L / (2 * L)

(540 / 480) = 1 / 2

Simplifying the equation, we find:

(540 / 480) = 1 / (2/1)

(540 / 480) = 1 / 2

Since the ratio of the frequencies is equal to the ratio of the lengths (540/480 = 1/2), we can conclude that the length of the pipe (L) must be half the length of the wavelength of the fundamental frequency, or L = λ/2.

Therefore, L = 240 cm, and the frequency of the fundamental (f1) is 480 Hz.

To summarize:
- The length of the pipe (L) is 240 cm
- The frequency of the fundamental (f1) is 480 Hz.