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Chapter 12: Problem 67

A sound wave arriving at your ear is transferred to the fluid in the cochlea. If the intensity in the fluid is 0.80 times that in air and the frequency is the same as for the wave in air, what will be the ratio of the pressure amplitude of the wave in air to that in the fluid? Approximate the fluid as having the same values of density and speed of sound as water.

Short Answer

Expert verified
Answer: The ratio of pressure amplitude of the wave in air to that in the fluid is approximately 16.04.

Step by step solution

01

Write down the given information

Here are the given details: 1. Intensity ratio (fluid to air) = 0.80 2. Frequency of air wave = Frequency of fluid wave 3. Fluid approximated as water (Same density and speed of sound)
02

Recall the formulas for Intensity and Pressure Amplitude

Intensity (I) is given by the formula: I = (P^2) / (2蟻v) Where, P is the pressure amplitude 蟻 is the density of the medium v is the speed of sound in the medium
03

Write down the formula for Intensity Ratios

Using the formula above, we can write the ratio of intensities for fluid and air as: (If/Ia) = (Pf^2)/(Pa^2) * (蟻av)/(蟻fv) Where, If is the Intensity in fluid Ia is the Intensity in air Pf is the Pressure Amplitude in fluid Pa is the Pressure Amplitude in air 蟻f is the density of fluid 蟻a is the density of air vf is the speed of sound in fluid va is the speed of sound in air
04

Use the given Intensity Ratio and plug in the values

We are given the intensity ratio If/Ia = 0.80. Insert it in the above equation: 0.80 = (Pf^2)/(Pa^2) * (蟻av)/(蟻fv) We need to find the ratio of pressure amplitude, which is (Pa/Pf): (Pa/Pf)^2 = (蟻fv)/(蟻av) / 0.80
05

Recall that fluid is approximately water

Since we approximate the fluid to have the same values of density and speed of sound as water, we use the standard values for water: 蟻f = 1000 kg/m^3 vf = 1500 m/s For air, we use the following standard values: 蟻a = 1.225 kg/m^3 va = 340 m/s Now plug these values into the equation: (Pa/Pf)^2 = ((1000)(1500))/((1.225)(340)) / 0.80
06

Calculate the ratio of Pressure Amplitudes

Calculate the result of the expression: (Pa/Pf)^2 = 257.4 Take the square root of this value to find the ratio of Pressure Amplitudes: Pa/Pf = sqrt(257.4) 鈮� 16.04 So, the ratio of pressure amplitude of the wave in air to that in the fluid is approximately 16.04.

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