Final answer:
The question involves finding two wave functions which interfere to form a given resultant wave function. By identifying the form of the equation, we can extract two complementary wave functions and verify their interference pattern by plotting them along with the resultant at a specific time.
Step-by-step explanation:
Interference of Wave Functions
The question at hand involves determining two wave functions that when combined produce a given resultant wave function. The given resultant wave function is YR (x, t) = 6.00 cm sin (3.00 m¯¹ x + 1.20 rad) cos (6.00 s¯¹t + 1.20 rad). We can extract two wave functions from this equation by recognizing that it has the form of the sum of two traveling waves as shown in an earlier example (YR = 2A sin(kx + β) cos (wt + β/2)), where A is the amplitude, k is the wave number, and w is the angular frequency. The two wave functions that interfere to form YR can be written as y1 (x, t) = A sin (kx - wt) and y2 (x, t) = A sin (kx + wt + π), with A equal to half the amplitude of YR (since YR is effectively doubled in amplitude), k corresponding to the wave number in the sine term, and w corresponding to the angular frequency in the cosine term.
Using A = 3 cm, k = 3.00 m¯¹, and w = 6.00 s¯¹, we can write our two wave functions as:
- y1 (x, t) = 3 cm sin (3.00 m¯¹ x - 6.00 s¯¹ t)
- y2 (x, t) = 3 cm sin (3.00 m¯¹ x + 6.00 s¯¹ t + 1.20 rad)
To verify that these functions combine to form the resultant wave YR, we can plot y1, y2, and YR at a given time, t = 1.00 s. The plot will show the individual wave forms and the resulting interference pattern when they are summed. The plot should illustrate that the interference of y1 and y2 indeed produces a wave pattern matching the given YR function.