Problem 2 - University Physics with Modern Physics

Question

Example 16.1 (Section 16.1\()\) showed that for sound waves in air with frequency 1000 \(\mathrm{Hz}\) , a displacement amplitude of \(1.2 \times 10^{-8} \mathrm{m}\) produces a pressure amplitude of \(3.0 \times 10^{-2}\) Pa. Water at \(20^{\circ} \mathrm{C}\) has a bulk modulus of \(2.2 \times 10^{9} \mathrm{Pa}\) and the speed of sound in water at this temperature is 1480 \(\mathrm{m} / \mathrm{s} .\) For \(1000-\mathrm{Hz}\) sound waves in \(20^{\circ} \mathrm{C}\) water, what displacement amplitude is produced if the pressure amplitude is \(3.0 \times 10^{-2}\) Pa? Explain whyyour answer is much less than \(1.2 \times 10^{-8} \mathrm{m}\) .

Step-by-Step Solution

Answer

The displacement amplitude in water is approximately 3.21 × 10^{-12} m, much smaller than in air due to water's higher bulk modulus.

Step 1: Understanding Given Values

We need to determine the displacement amplitude for sound waves in water where the frequency is 1000 Hz, the pressure amplitude is 3.0 × 10^{-2} Pa, the bulk modulus is 2.2 × 10^9 Pa, and the speed of sound is 1480 m/s. Compare this with air, where the displacement amplitude was 1.2 × 10^{-8} m for the same pressure amplitude.

Step 2: Formula for Displacement Amplitude

The formula for the displacement amplitude (\( s_0 \) ) is given by:\[s_0 = \frac{\Delta p}{B \cdot k}\]where \( \Delta p \) is the pressure amplitude, \( B \) is the bulk modulus, and \( k \) is the wave number.

Step 3: Calculate the Wave Number

The wave number (\( k \) ) is determined by the equation:\[k = \frac{2\pi f}{v}\]where \( f \) is the frequency and \( v \) is the speed of sound. Substituting the given values:\[k = \frac{2\pi \times 1000}{1480} \approx 4.25 \text{ m}^{-1}\]

Step 4: Compute the Displacement Amplitude

Now, use the earlier calculated wave number and given values:\[s_0 = \frac{3.0 \times 10^{-2}}{2.2 \times 10^9 \times 4.25} \approx 3.21 \times 10^{-12} \text{ m}\]

Step 5: Explain the Small Displacement Amplitude

Water is much less compressible than air due to its higher bulk modulus, which requires a much smaller displacement amplitude to achieve the same pressure amplitude. This means the sound wave in water causes less physical displacement of particles compared to air under similar conditions.

Key Concepts

Sound Waves

Sound waves are a type of mechanical wave that travels through a medium (like air or water) by compressing and rarefying the particles in the medium. These waves carry energy from one point to another and are understood as oscillations created by vibrating sources, such as a speaker or a musical instrument.
Unlike electromagnetic waves, sound waves require a material medium to travel. They can sweep through gases, liquids, and solids, and their speed varies depending on the density and elasticity of the medium. This is why sound travels faster in water than in air because water molecules are closer together and thus transmit vibrations more efficiently.

• Direction of oscillation is parallel to the direction of wave travel.
• They are longitudinal waves, meaning the oscillation occurs in the same direction as the wave movement.
• Sound wave speed depends on properties of the medium, like density and bulk modulus.

This basic understanding of sound waves forms the foundation for calculating related parameters like displacement amplitude.

Pressure Amplitude

Pressure amplitude in sound waves is the variation in pressure caused by the wave relative to the ambient atmospheric pressure. It is crucial for quantifying how intense a sound wave is. The higher the pressure amplitude, the louder the sound is perceived.
When a sound wave passes through a medium, it alternates between compressions and rarefactions, creating regions of high and low pressure. The difference between the maximum and minimum pressure levels defines the pressure amplitude.

• Pressure amplitude is directly linked to the perceived volume of sound.
• Measured in Pascals (Pa).
• Higher pressure amplitude indicates stronger sound waves.

The concept is important for understanding why the displacement amplitude in water is significantly smaller than in air for the same pressure amplitude. Since water is less compressible, it easily transmits sound pressure with minimal particle displacement.

Bulk Modulus

Bulk modulus is a property of a material that indicates how compressible that material is. Essentially, it measures the material's ability to withstand changes in volume when under pressure. For example, water has a bulk modulus of about \(2.2 \times 10^9\) Pa, suggesting it is quite incompressible relative to air.
Mathematically, bulk modulus \( B \) is described as the ratio between the change in pressure \( \Delta P \) and the relative change in volume \( (\frac{\Delta V}{V}) \).
\[ B = - V\frac{\Delta P}{\Delta V} \]

• High bulk modulus means less compressibility.
• Lower displacement amplitude for the same pressure amplitude.
• Important for calculating sound speed in materials.

Knowing the bulk modulus allows us to understand displacement amplitude changes under different conditions. More solid materials with higher moduli transmit sound differently than gases, evident in the difference in displacement amplitudes between water and air.

Speed of Sound

The speed of sound is determined by how fast sound waves can pass through a material. It is influenced significantly by the properties of the medium: density and elasticity (linked to bulk modulus). For water at \(20^{\circ}C\), the speed of sound is approximately \(1480\) m/s, much greater than in air.

The speed of sound \( v \) in a medium is mathematically given by:\[ v = \sqrt{\frac{B}{\rho}} \] where \( B \) is the bulk modulus and \( \rho \) is the density of the material.

• Depends on medium properties: mostly bulk modulus and density.
• Greater in water due to high bulk modulus and moderate density.
• Affects the way sound is transmitted, impacted by temperature.

Understanding the speed of sound helps in determining wave-related quantities such as frequency and wave number, which are vital in finding displacement amplitudes for sound waves in different media.