Question Details:
A row of seats is parallel to a stage at a distance of 8.70 m from it. At the center and front of the stage is a diffraction horn loudspeaker that has a width of 7.50 cm. The speaker is playing a tone at a frequency of 9300 Hz . The speed of sound is 344 m/s.
What is the separation between two seats, located on opposite sides of the center of the row, at which the tone cannot be heard?
I divided 344 by 9300 and got 0.03699
and then i drew a triangle and divided 7.50 by 3.6989 which gave me 2.02 i then use the pygathorean theorem, and got the additional side as 6.52 which divided 3.6989, and then i used that value to multiply vy 8.70m to get the distance from the centre.
I am stuck can someone please help. I keep getting the wrong answer but I don't understand why.
Minimum intensity occurs at an angle θmin given by dsinθmin = λ
d is the slit width (0.075m)
Stadnik's class?
yeah i did that but the answer is incorrect
To solve this problem, let's go through the steps one by one:
Step 1: Calculate the wavelength (λ) of the sound wave.
The speed of sound (v) is given as 344 m/s, and the frequency (f) is 9300 Hz.
The formula to calculate wavelength is:
λ = v / f
λ = 344 m/s / 9300 Hz
Convert Hz to s⁻¹ by dividing by 1/s:
λ = 344 m/s / 9300 s⁻¹
λ ≈ 0.037 m
Step 2: Calculate the angle (θ) of diffraction.
The speaker has a width of 7.50 cm, which is the same as 0.075 m.
The formula to calculate the angle of diffraction is:
θ ≈ λ / w
θ ≈ 0.037 m / 0.075 m
θ ≈ 0.493 rad
Step 3: Calculate the distance (d) from the center where the tone cannot be heard.
At the center, the speaker is playing towards the opposite side, so we can consider the center as a point source and use the formula for the single slit diffraction pattern:
d ≈ (w / 2) / tan(θ/2)
d ≈ (0.075 m / 2) / tan(0.493 rad / 2)
d ≈ 0.0375 m / tan(0.2465 rad)
d ≈ 0.0375 m / 0.2532
d ≈ 0.1481 m
Step 4: Multiply the distance (d) by 2 to get the separation between two seats.
The seats are on opposite sides of the center, so the total separation is twice the distance (d).
Separation = 2 * 0.1481 m
Separation ≈ 0.2962 m
Therefore, the separation between two seats, located on opposite sides of the center of the row, at which the tone cannot be heard, is approximately 0.2962 meters.
To solve this problem, we need to consider the diffraction of sound waves around the speaker. Diffraction is a phenomenon that occurs when waves encounter obstacles or pass through openings and bend around them.
First, let's calculate the wavelength of the sound wave. The wavelength (λ) can be found using the formula:
λ = v/f
where v is the speed of sound and f is the frequency of the tone. Substituting the given values:
λ = 344 m/s / 9300 Hz
λ ≈ 0.037 m
Next, let's consider diffraction. According to the principle of diffraction, sound waves can diffract around an obstacle if the dimensions of the obstacle are comparable to or smaller than the wavelength.
In this case, the width of the diffraction horn speaker (7.50 cm) is comparable to the wavelength of the sound wave (0.037 m). Therefore, we can expect diffraction to occur.
To find the separation between two seats at which the tone cannot be heard, we need to consider the phenomenon called the first minimum. In diffraction, the first minimum is the angle at which the diffracted waves cancel out each other, resulting in no sound being heard.
To calculate the angle at which the first minimum occurs, we can use the formula:
sin(θ) = λ / w
where θ is the angle, λ is the wavelength, and w is the width of the speaker.
Substituting the values:
sin(θ) = 0.037 m / 0.075 m = 0.4933
To find θ, we take the inverse sine (sin^(-1)) of 0.4933:
θ ≈ sin^(-1)(0.4933) ≈ 29.7 degrees
Now we can find the distance from the center of the row to the point where the tone cannot be heard. Let's call this distance d.
Using trigonometry, we can set up the equation:
tan(θ) = d / distance from the center to the stage
tan(29.7 degrees) = d / 8.70 m
Solving for d:
d ≈ tan(29.7 degrees) * 8.70 m
d ≈ 5.2297 m
Therefore, the separation between two seats at which the tone cannot be heard is approximately 5.23 meters.