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?
Please explain the meaning of ö1 and ö2. Are they phase angles?
Your /5 rad number does not make sense. Where is the rest of the fraction?
What it the equation for displacement for a single wave? Are ym1 and ym2 individual wave amplitudes?
For two waves with the same frequency, the combined wave has the same frequency and an amplitude that is the vector sum of the two phasor amplitudes.
To find the amplitude of the resultant wave, we can use the principle of superposition for waves:
The principle of superposition states that when two or more waves meet at a point in space, the resultant displacement at that point is equal to the algebraic sum of the individual displacements.
In this case, we have two sinusoidal waves with the same frequency and traveling in the same direction along a string. Let's denote the amplitudes of the two waves as A1 and A2, and the phase angles as ø1 and ø2, respectively.
The general equation for a sinusoidal wave is given by:
y = A * sin(ωt + ø),
where y represents the displacement of the wave at a particular point and time, A is the amplitude, ω is the angular frequency, t is time, and ø is the phase angle.
In this case, both waves have the same frequency, so ω is the same for both. Let's denote it as ω.
The displacement function for the first wave is given by:
y1 = A1 * sin(ωt + ø1),
and the displacement function for the second wave is given by:
y2 = A2 * sin(ωt + ø2).
Now, let's find the resultant wave. The displacement of the resultant wave, y, is given by:
y = y1 + y2.
Substituting the expressions for y1 and y2, we have:
y = A1 * sin(ωt + ø1) + A2 * sin(ωt + ø2).
To simplify this expression, we can use the trigonometric identity:
sin(a + b) = sin(a) * cos(b) + cos(a) * sin(b).
Applying this identity, we get:
y = (A1 * cos(ø1) + A2 * cos(ø2)) * sin(ωt) + (A1 * sin(ø1) + A2 * sin(ø2)) * cos(ωt).
Now, let's rewrite this equation in the form:
y = A * sin(ωt + ø).
Comparing coefficients, we can find the amplitude, A, and the phase angle, ø, of the resultant wave.
We have:
A * sin(ωt + ø) = (A1 * cos(ø1) + A2 * cos(ø2)) * sin(ωt) + (A1 * sin(ø1) + A2 * sin(ø2)) * cos(ωt).
Comparing the terms on both sides, we get:
A * sin(ø) = A1 * cos(ø1) + A2 * cos(ø2),
and
A * cos(ø) = A1 * sin(ø1) + A2 * sin(ø2).
To solve for A, we can square both equations and add them:
(A * sin(ø))^2 + (A * cos(ø))^2 = (A1 * cos(ø1) + A2 * cos(ø2))^2 + (A1 * sin(ø1) + A2 * sin(ø2))^2.
Simplifying the right-hand side, we have:
A^2 * (sin^2(ø) + cos^2(ø)) = A1^2 + 2 * A1 * A2 * cos(ø1 - ø2) + A2^2.
Using the trigonometric identity for sin^2(ø) + cos^2(ø), which is equal to 1, we can rewrite the equation as:
A^2 = A1^2 + 2 * A1 * A2 * cos(ø1 - ø2) + A2^2.
Finally, taking the square root of both sides, we get:
A = sqrt(A1^2 + 2 * A1 * A2 * cos(ø1 - ø2) + A2^2).
Now let's substitute the given values:
A = sqrt(4.2^2 + 2 * 4.2 * 5.9 * cos(0 - π/5) + 5.9^2).
Calculating this expression, we find:
A ≈ 6.59 cm.
Therefore, the amplitude of the resultant wave is approximately 6.59 cm.