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Sinusoidal waves are fundamental in understanding wave behavior because they model many natural phenomena, like sound and light waves. A sinusoidal wave is a mathematical curve describing a smooth periodic oscillation, characterized by its amplitude, wavelength, frequency, and phase.

• Amplitude (\( y_m\) ) : This defines the maximum displacement from the wave's rest position.
• Wavelength (\( \lambda\) ) : It is the distance over which the wave's shape repeats.
• Frequency (\( f\) ) : This tells us how many waves pass through a point in a given time period.
• Phase (\( \phi\) ) : This denotes the horizontal shift of the wave.

When added together, multiple sinusoidal waves can produce a resultant wave that is significantly different from the individual waves. The wave equation for a sinusoidal wave is typically expressed as: \[y = y_m \sin(kx - \omega t + \phi)\]where \( k \) is the wave number and \( \omega \) is the angular frequency. Understanding each component is essential for applying concepts like the superposition principle, which allows us to predict how different waves interact.

Calculating the amplitude of the resultant wave from two overlapping sinusoidal waves involves using the Superposition Principle. This principle states that the resultant wave is the sum of the displacements due to each of the individual waves.
To find the amplitude, we use the formula:

• \[ y_{mr} = \sqrt{y_{m1}^2 + y_{m2}^2 + 2y_{m1}y_{m2} \cos(\phi_2 - \phi_1)} \]

This formula considers not only the individual amplitudes \( y_{m1} \) and \( y_{m2} \) but also the phase difference (\( \phi_2 - \phi_1 \) ) between the waves.
Substitute the values:

• For wave 1, \( y_{m1} = 3.0 \, \text{mm} \), \( \phi_1 = 0 \)
• For wave 2, \( y_{m2} = 5.0 \, \text{mm} \), \( \phi_2 = 70^{\circ} \)

Calculate the amplitude:\[ y_{mr} = \sqrt{3^2 + 5^2 + 2 \times 3 \times 5 \times \cos(70^{\circ})} \approx 7.5 \, \text{mm} \]This result shows how overlapping sinusoidal waves can combine to form a larger amplitude wave through constructive interference.

The phase constant of a wave helps to determine the initial angle or position of the wave. When two sinusoidal waves combine, the phase constant of the resultant wave indicates how the overlap affects the wave's oscillation in time and space.
To determine the phase constant \( \phi_r \) of the resultant wave from two overlapping waves, use the formula:

• \[\tan(\phi_r) = \frac{y_{m1} \sin(\phi_1) + y_{m2} \sin(\phi_2)}{y_{m1} \cos(\phi_1) + y_{m2} \cos(\phi_2)}\]

For our situation:

• Since \( \phi_1 = 0 \), \( \sin(0) = 0 \) and \( \cos(0) = 1 \)
• Plug in values:\[\tan(\phi_r) = \frac{5 \times \sin(70^{\circ})}{3 + 5 \times \cos(70^{\circ})}\]
• Simplify:\[\tan(\phi_r) \approx \frac{4.7}{4.71} \]

Thus, \( \phi_r \approx \tan^{-1}(0.998) \), which is approximately \( 44.9^{\circ} \).
This calculation shows the phase relationship between combined waves, illustrating both how the waves shift each other and the timing of their peaks and troughs.