Understanding the speed of waves is crucial in physics, particularly when distinguishing between transverse and longitudinal waves. While transverse waves travel along strings, longitudinal waves propagate through various mediums, including liquids and solids. The equations governing the speed of these waves differ slightly based on the medium in which they travel.
For longitudinal waves in fluids, the speed can be expressed with the formula:
$$ v = \sqrt{\frac{\beta}{\rho}} $$
where \( v \) is the wave speed, \( \beta \) is the bulk modulus of the fluid, and \( \rho \) is the density of the fluid. The bulk modulus is a measure of a substance's resistance to uniform compression and is typically provided in problems involving fluid dynamics.
In contrast, for longitudinal waves in solids, the equation is similar but utilizes Young's modulus instead:
$$ v = \sqrt{\frac{E}{\rho}} $$
where \( E \) represents Young's modulus, a measure of the stiffness of a solid material. Both equations highlight that the speed of waves is influenced by the material properties of the medium.
Regardless of the type of wave, the relationship between wave speed, wavelength (\( \lambda \)), and frequency (\( f \)) remains consistent:
$$ v = \lambda \cdot f $$
To illustrate these concepts, consider a liquid with a density of 1200 kg/m³, a frequency of 400 Hz, and a wavelength of 8 meters. To find the bulk modulus (\( \beta \)), we can rearrange the wave speed equation:
1. Calculate wave speed: $$ v = \lambda \cdot f = 8 \, \text{m} \cdot 400 \, \text{Hz} = 3200 \, \text{m/s} $$
2. Substitute into the bulk modulus equation: $$ \sqrt{\frac{\beta}{1200}} = 3200 $$
3. Squaring both sides gives: $$ \frac{\beta}{1200} = 10240000 $$
4. Thus, $$ \beta = 1.23 \times 10^{10} \, \text{Pascals} $$
In another example, consider a 60-meter-long brass rod. To determine the time (\( \Delta t \)) it takes for sound to travel through the rod after being struck, we can use the relationship between distance, speed, and time:
1. First, calculate the speed of sound in brass using Young's modulus and density:
$$ v = \sqrt{\frac{9 \times 10^{10}}{86100}} \approx 3200 \, \text{m/s} $$
2. Then, apply the formula for time: $$ \Delta t = \frac{\Delta x}{v} = \frac{60 \, \text{m}}{3200 \, \text{m/s}} = 0.01875 \, \text{s} \approx 0.02 \, \text{s} $$
This example illustrates that sound travels significantly faster through solids like brass compared to air, due to the higher density and stiffness of the material. Understanding these principles is essential for solving problems related to wave propagation in different mediums.
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