Two sinusoidal waves of the same frequency travel in the same direction along a string. If ym1 = 4.2 cm, ym2 = 5.9 cm, ö1 = 0, and ö2 = /5 rad, what is the amplitude (in cm) of the resultant wave?
To find the amplitude of the resultant wave, we need to use the concept of superposition.
Superposition states that when two waves with the same frequency overlap, their displacements add together at each point. The resulting wave at any instant is the sum of the displacements of the individual waves.
In this case, we have two sinusoidal waves with the same frequency traveling in the same direction along a string. Let's denote the amplitudes of these waves as A1 and A2, and the phase angles as ö1 and ö2.
The equation for a sinusoidal wave can be written as y = A * sin(ωt + ö), where y is the displacement, A is the amplitude, ω is the angular frequency, t is the time, and ö is the phase angle.
Given that ym1 = 4.2 cm, ym2 = 5.9 cm, ö1 = 0, and ö2 = π/5 rad, we can write the equations for the individual waves as:
y1 = A1 * sin(ωt)
y2 = A2 * sin(ωt + π/5)
To find the amplitude of the resultant wave, we need to add the displacements of the individual waves at every point. Using the principle of superposition, we have:
y_resultant = y1 + y2
= A1 * sin(ωt) + A2 * sin(ωt + π/5)
We can rewrite the equation by applying the trigonometric identity for the sum of sines:
y_resultant = (A1 * cos(π/5) + A2) * sin(ωt) + A1 * sin(π/5) * cos(ωt)
The resulting wave has the form y_resultant = A_resultant * sin(ωt), where A_resultant is the amplitude of the resultant wave. By comparing this equation with the equation for the resulting wave, we can find the amplitude as:
A_resultant = √((A1 * cos(π/5) + A2)^2 + (A1 * sin(π/5))^2)
Substituting the given values, we have:
A_resultant = √((4.2 * cos(π/5) + 5.9)^2 + (4.2 * sin(π/5))^2)
Evaluating this expression will give you the amplitude of the resultant wave in cm.