Two sinusoidal waves are described by the equations y1=(5.00...

Two sinusoidal waves are described by the equations $$y_{1}=(5.00 \mathrm{~m}) \sin [\pi(4.00 x-1200 t)]$$ and $$y_{2}=(5.00 \mathrm{~m}) \sin [\pi(4.00 x-1200 t-0.250)]$$ where $x, y_{1}$, and $y_{2}$ are in meters and $t$ is in seconds. (a) What is the amplitude of the resultant wave? (b) What is the frequency of the resultant wave?

Two sinusoidal waves are described by the equations
$$y_{1}=(5.00 \mathrm{~m}) \sin [\pi(4.00 x-1200 t)]$$
and
$$y_{2}=(5.00 \mathrm{~m}) \sin [\pi(4.00 x-1200 t-0.250)]$$
where $x, y_{1}$, and $y_{2}$ are in meters and $t$ is in seconds.
(a) What is the amplitude of the resultant wave?
(b) What is the frequency of the resultant wave?

Key Concepts

Resultant Amplitude from Phase Addition

When two sinusoidal waves of identical frequency and amplitude interfere, their resultant amplitude can be determined by vector or phasor addition. The overall amplitude depends on the cosine of half the phase difference between the waves, illustrating how even slight differences in phase can significantly alter the combined wave’s amplitude.

Superposition Principle

The superposition principle states that when two or more waves overlap in space, the resulting wave displacement is the algebraic sum of the displacements of the individual waves. This concept is fundamental to understanding how complex wave patterns form from simpler sinusoidal functions.

Phase Difference

Phase difference refers to the offset in the initial angles of two sinusoidal waves, which determines the extent to which their peaks, troughs, and zero crossings align. The phase difference is crucial in determining whether the interference between waves is constructive, destructive, or somewhere in between, thereby affecting the resultant amplitude.

Interference of Waves

Interference describes the phenomenon that occurs when two waves combine to form a new wave pattern. Depending on their phase relationship, the interference can enhance the amplitude (constructive interference) or reduce it (destructive interference). This concept is key to understanding how the amplitudes of multiple waves add together.

Angular Frequency and Frequency

Angular frequency is a measure of how rapidly a wave oscillates in terms of radians per second, and it is related to the frequency (cycles per second) by the factor 2?. Understanding this relationship is important when working with sinusoidal wave equations, as it allows for conversions between the mathematical form of the wave and its physical interpretation.

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