When two sinusoidal waves of equal amplitude are out of phase, the resultant wave is obtained by adding the individual waves together at each point in time. To determine the amplitude of the resultant wave, we need to consider the phase relationship between the waves.
If the two waves are 1/4 wavelength out of phase, it means that there is a phase difference of π/2 radians (or 90 degrees) between them. This corresponds to a phase shift of one-fourth of a complete cycle.
When two waves with the same frequency and amplitude are added together, the resultant amplitude can be found using the concept of vector addition. In this case, since the waves are out of phase by π/2 radians, their vector components are perpendicular to each other.
When two vectors of equal magnitude are perpendicular to each other, the magnitude of their resultant vector (obtained by vector addition) is given by the formula:
Resultant Amplitude = √(Amplitude^2 + Amplitude^2)
Simplifying the equation, we get:
Resultant Amplitude = √(2 * Amplitude^2)
Resultant Amplitude = √2 * Amplitude
Therefore, the amplitude of the resultant wave when two sinusoidal waves of equal amplitude are 1/4 wavelength out of phase is √2 times the amplitude of each individual wave.