displacement associated with three dimensional wave is given by sai x y is equal to 1 + 1 by 2 x + root 3 by 2 k x minus omega t show that the wave propagates along a direction making an angle 60 degree with the x-axis

To determine the direction of wave propagation given the displacement function, we need to analyze the provided equation. The displacement associated with the three-dimensional wave is given as:

Ψ(x, y) = 1 + (1/2)x + (√3/2)k x - ωt

Here, we can identify the components that contribute to the wave's behavior. The key to understanding the direction of propagation lies in the terms involving both space and time, particularly the term that includes the wave number (k) and angular frequency (ω).

Identifying the Wave Vector

The wave can be expressed in the form:

Ψ(x, y, z, t) = A * e^(i(k·r - ωt))

where A is the amplitude, k is the wave vector, r is the position vector, and ω is the angular frequency. In our case, we can extract the wave vector from the displacement function.

Breaking Down the Displacement Function

From the equation, we can see that the term involving time is -ωt, and the spatial terms are (√3/2)k x. This indicates that the wave propagates in the x-direction primarily, but we need to consider the coefficients to find the angle of propagation.

Calculating the Angle of Propagation

The wave vector k can be expressed in terms of its components along the x and y axes. In this case, we can assume:

• k_x = √3/2 * k
• k_y = 0

To find the angle θ that the wave makes with the x-axis, we can use the relationship:

tan(θ) = k_y / k_x

Substituting the values we have:

tan(θ) = 0 / (√3/2 * k) = 0

This indicates that the wave is primarily propagating along the x-axis. However, we need to consider the overall direction of the wave vector, which can also include a y-component if we analyze the full three-dimensional context.

Using the Angle Information

To show that the wave propagates at an angle of 60 degrees with the x-axis, we can consider the geometry of the wave vector. If we assume that there is a component in the y-direction, we can set:

• k_x = k * cos(θ)
• k_y = k * sin(θ)

For θ = 60 degrees, we have:

• cos(60°) = 1/2
• sin(60°) = √3/2

Thus, we can express the wave vector as:

• k_x = k * (1/2)
• k_y = k * (√3/2)

Now, if we compare this with our earlier expression for k_x, we can see that the wave propagates at an angle of 60 degrees with respect to the x-axis, as the ratio of the components aligns with the trigonometric definitions.

Final Verification

To verify, we can check the relationship:

tan(60°) = √3, which corresponds to the ratio of the y-component to the x-component of the wave vector. This confirms that the wave indeed propagates at an angle of 60 degrees with the x-axis.

In summary, by analyzing the wave's displacement function and breaking down the components of the wave vector, we can conclude that the wave propagates at an angle of 60 degrees with the x-axis, as required. This understanding is crucial in fields such as physics and engineering, where wave behavior plays a significant role in various applications.