In the study of wave interference, the principle of superposition plays a crucial role in understanding how multiple wave functions combine to create a resultant wave. This principle states that when two or more waves overlap, the total displacement at any point is simply the sum of the individual displacements of the waves. This can be expressed mathematically as:
$$y_{\text{net}} = y_1 + y_2$$
Where \(y_1\) and \(y_2\) are the individual wave functions. These wave functions can be represented as sine or cosine functions, and they do not need to have the same amplitude, wave number, or frequency. For example, if we have:
$$y_1 = A_1 \sin(k_1 x \pm \omega_1 t)$$
$$y_2 = A_2 \cos(k_2 x \pm \omega_2 t)$$
Then the net wave function becomes:
$$y_{\text{net}} = A_1 \sin(k_1 x \pm \omega_1 t) + A_2 \cos(k_2 x \pm \omega_2 t)$$
To find the displacement of a particle at a specific position \(x\) and time \(t\), you simply substitute these values into the wave functions. For instance, if we have the wave functions:
$$y_1 = 0.3 \sin(4x - 1.6t)$$
$$y_2 = 0.7 \cos(5x - 2t)$$
To calculate the displacement at \(x = 2\) and \(t = 0.5\), we substitute these values into the wave functions:
$$y_{\text{net}} = 0.3 \sin(4(2) - 1.6(0.5)) + 0.7 \cos(5(2) - 2(0.5))$$
Calculating each term separately, we find:
1. For the sine function:
$$0.3 \sin(8 - 0.8) = 0.3 \sin(7.2) \approx 0.24$$
2. For the cosine function:
$$0.7 \cos(10 - 1) = 0.7 \cos(9) \approx -0.64$$
Finally, the net displacement is:
$$y_{\text{net}} = 0.24 - 0.64 = -0.4$$
This result indicates the overall displacement of the particle due to the interference of the two waves. Understanding this process is essential for analyzing wave behavior in various physical contexts, such as sound and light waves.
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