A 3.00 kHz tone is being produced by a speaker with a diameter of 0.175 m. The air temperature changes from 0 to 29°C. Assuming air to be an ideal gas, find the change in the diffraction angle
My solution
v directly proportional to sqrt T
v = sqrt 302 =17.382
lambda = v/f = 17.382/3000
=0.005794
Sin theta = 1.22 lambda/ diameter
= 1.22x 0.005794/0.175
=0.009425
theta = Sin^-1 0.009425
= 0.54
answer is wrong
First, v is prop to sqrt T does NOT mean that v=sqrt(Temp). That is plain wrong. So lambda that follows is wrong, and the angle also.
http://www.sengpielaudio.com/calculator-speedsound.htm
I used the calculator but stll got the wrong answer. PLease show work
To find the change in the diffraction angle, we need to use the correct relationship between the speed of sound and temperature.
The relationship between the speed of sound and temperature in an ideal gas is given by:
v = sqrt(gamma * R * T)
where:
- v is the speed of sound
- gamma is the adiabatic index (a constant that depends on the gas)
- R is the gas constant
- T is the temperature in Kelvin
First, we need to convert the temperature in degrees Celsius to Kelvin. We can do this by adding 273.15 to the temperature in Celsius:
T1 = 0 + 273.15 = 273.15 K (initial temperature)
T2 = 29 + 273.15 = 302.15 K (final temperature)
Next, we need to determine the adiabatic index, gamma. For air, gamma is approximately 1.4.
Now we can calculate the initial speed of sound:
v1 = sqrt(gamma * R * T1)
The gas constant, R, for air is approximately 287 J/(kg·K).
Substituting the values, we have:
v1 = sqrt(1.4 * 287 * 273.15)
Calculating v1, we find:
v1 ≈ 331.4 m/s
Next, we can calculate the final speed of sound:
v2 = sqrt(gamma * R * T2)
Substituting the values, we have:
v2 = sqrt(1.4 * 287 * 302.15)
Calculating v2, we find:
v2 ≈ 346.9 m/s
Now we can calculate the change in the diffraction angle:
Δθ = sin^(-1)((1.22 * λ) / D)
where:
- Δθ is the change in the diffraction angle
- λ is the wavelength of the sound wave
- D is the diameter of the speaker
To find the wavelength, we use the formula:
λ = v / f
where:
- v is the speed of sound
- f is the frequency of the sound wave
The given frequency is 3.00 kHz, which is equivalent to 3000 Hz.
Using the initial speed of sound, we have:
λ1 = v1 / f
Substituting the values, we have:
λ1 = 331.4 / 3000
Calculating λ1, we find:
λ1 ≈ 0.1105 m
Similarly, using the final speed of sound, we have:
λ2 = v2 / f
Substituting the values, we have:
λ2 = 346.9 / 3000
Calculating λ2, we find:
λ2 ≈ 0.1156 m
Finally, substituting the values of λ1, λ2, and the diameter of the speaker (0.175 m) into the equation for the change in the diffraction angle, we can calculate Δθ:
Δθ ≈ sin^(-1)((1.22 * (λ2 - λ1)) / D)
Substituting the values, we have:
Δθ ≈ sin^(-1)((1.22 * (0.1156 - 0.1105)) / 0.175)
Calculating Δθ, we find:
Δθ ≈ sin^(-1)(0.0293)
Using a scientific calculator or an online calculator, we can find the inverse sine of 0.0293, which gives us:
Δθ ≈ 1.675 degrees
So the change in the diffraction angle is approximately 1.675 degrees.