To find the amplitudes of the two sound waves \( Y_1 \) and \( Y_2 \) at the specified values of \( x \) and \( t \), we can simply substitute these values into the equations provided. The amplitude of a wave is the coefficient \( A \) in the wave equation, which represents the maximum displacement from the equilibrium position. Let's break this down step by step.
Wave Equations
The equations for the two waves are given as:
- Wave 1: \( Y_1 = A \sin(0.5\pi x - 100\pi t) \)
- Wave 2: \( Y_2 = A \sin(0.48\pi x - 96\pi t) \)
Identifying the Amplitude
In both equations, \( A \) represents the amplitude of the waves. However, the exact value of \( A \) is not provided in the question. Therefore, we can only express the amplitudes in terms of \( A \).
Calculating the Values at \( x = 0 \) and \( t = 0.25 \)
Now, let's substitute \( x = 0 \) and \( t = 0.25 \) into both equations to find the values of \( Y_1 \) and \( Y_2 \).
For Wave 1: \( Y_1 \)
Substituting \( x = 0 \) and \( t = 0.25 \):
- Calculate \( Y_1 \):
- \( Y_1 = A \sin(0.5\pi(0) - 100\pi(0.25)) \)
- \( Y_1 = A \sin(0 - 25\pi) \)
- Since \( \sin(25\pi) = 0 \), we have \( Y_1 = A \cdot 0 = 0 \).
For Wave 2: \( Y_2 \)
Now, substituting the same values into the second wave:
- Calculate \( Y_2 \):
- \( Y_2 = A \sin(0.48\pi(0) - 96\pi(0.25)) \)
- \( Y_2 = A \sin(0 - 24\pi) \)
- Again, \( \sin(24\pi) = 0 \), so \( Y_2 = A \cdot 0 = 0 \).
Final Results
At the specified values of \( x = 0 \) and \( t = 0.25 \), both waves have a value of zero:
- Amplitude of \( Y_1 \): 0
- Amplitude of \( Y_2 \): 0
In summary, while the amplitudes of the waves are represented by \( A \), at the given points in time and space, both waves have a displacement of zero. This means that at \( x = 0 \) and \( t = 0.25 \), neither wave is at its maximum displacement; they are both at equilibrium. If you have any further questions about wave behavior or need clarification on any concepts, feel free to ask!
