Question

Two waves 1 and 2 are present on a string:
(35mm) sin [(8.4m ) (15.7s ) ] 1 1
y1 x t
− − = −
(35mm) sin [(8.4m ) (15.7s ) ] 1 1
y2 x t
− − = +
(i) Write the expression for the resultant wave, y = y1 + y2 in the form of wave function
for a standing wave. (ii) Determine the x coordinates of the first two antinodes, starting at
the origin and progressing towards + x direction. (iii) Determine the x coordinate of the
node that is between the antinodes of part (ii).

Answer 1

Answer on Question #40982, Physics, Mechanics | Kinematics | Dynamics

Two waves 1 and 2 are present on a string:

$$y1 = (35 \text{ mm}) \sin[(8.4m^{\wedge} - 1)x - (15.7s^{\wedge} - 1)t]$$ $$y2 = (35 \text{ mm}) \sin[(8.4m^{\wedge} - 1)x + (15.7s^{\wedge} - 1)t]$$

(i) Write the expression for the resultant wave, $y = y1 + y2$ in the form of wave function for a standing wave.

(ii) Determine the $x$ coordinates of the first two antinodes, starting at the origin and progressing towards $+$ $x$ direction.

(iii) Determine the $x$ coordinate of the node that is between the antinodes of part (ii).

Solution

(i) Two waves 1 and 2 are present on a string:

$$y_1 = 35 \sin(8.4x - 15.7t),$$ $$y_2 = 35 \sin(8.4x + 15.7t).$$

The sum of these two waves is:

$$y = y_1 + y_2 = 35 \sin(8.4x - 15.7t) + 35 \sin(8.4x + 15.7t).$$

We can use formulae for the sum of sinuses:

$$\sin a + \sin b = 2 \sin \frac{a + b}{2} \cos \frac{a - b}{2}.$$

So

$$y = y_1 + y_2 = 70 \sin(8.4x) \cos(15.7t).$$

(ii) The positions of the antinodes are given by

$$x = \frac{\lambda}{4}, \frac{3\lambda}{4}, \frac{5\lambda}{4}, \dots, \frac{(2n - 1)\lambda}{4}, n = 1,2,3 \dots$$

In our case $k = \frac{2\pi}{\lambda} = 8.4 \frac{rad}{m}$ and the wavelength is $\lambda = \frac{2\pi}{8.4} = 0.748 \, m$.

The $x$ coordinates of the first two antinodes are

$$x_1 = \frac{\lambda}{4} = 0.187 \, m,$$ $$x_2 = \frac{3\lambda}{4} = 0.561 \, m.$$

(iii) The node is located at

$$x = \frac{\lambda}{2} = 0.374 \, m.$$

Answer: (i) $70 \sin(8.4x) \cos(15.7t)$; (ii) $0.187 \, m$, $0.561 \, m$; (iii) $0.374 \, m$.