Two waves traveling together along the same line are given...

Two waves traveling together along the same line are given by $y_{1}=5 \sin (\omega t+\pi / 2)$ and $y_{2}=7 \sin (\omega t+\pi / 3)$. Find $(a)$ the resultant amplitude, (b) the initial phase angle of the resultant, and ( $c$ ) the resultant equation of motion. Ans. $(a) 11.60,(b) 72.4^{\circ},(c) y=11.60 \sin \left(\omega t+72.4^{\circ}\right)$

Two waves traveling together along the same line are given by $y_{1}=5 \sin (\omega t+\pi / 2)$ and $y_{2}=7 \sin (\omega t+\pi / 3)$. Find $(a)$ the resultant amplitude, (b) the initial phase angle of the resultant, and ( $c$ ) the resultant equation of motion.
Ans. $(a) 11.60,(b) 72.4^{\circ},(c) y=11.60 \sin \left(\omega t+72.4^{\circ}\right)$

Key Concepts

Vector Addition of Sinusoids

When two sinusoidal waves of the same frequency are superimposed, they can be combined by adding their corresponding phasors as vectors. The resultant vector's magnitude represents the amplitude of the new wave, while its direction (angle) gives the phase shift. This method provides a clear geometric interpretation of the resultant amplitude and phase.

Resultant Sinusoidal Wave

The outcome of adding two sinusoidal waves of the same frequency is another sinusoid. This resultant wave can be expressed in a form that directly shows its amplitude and phase shift, encapsulating the entire summation process into a single, coherent equation of motion.

Phasor Representation

Phasor representation is a method of expressing sinusoidal functions as rotating vectors (phasors) in the complex plane. Each sinusoid is represented by a vector whose length corresponds to the amplitude and whose angle corresponds to the phase. This representation greatly simplifies the process of adding sinusoids with the same angular frequency.

Superposition Principle

This concept refers to the idea that when two or more waves travel through the same medium, their displacements add algebraically at each point. The overall effect or resultant wave is obtained by summing the individual waves, which is a fundamental property of linear systems and is essential in understanding phenomena such as interference and beats.

Transcript

00:01 So in the question we are given two waves.

00:05 We are given pi 1 which is 5 7 omega t plus pi by 2 and also we are given another wave by 2 which is 7 sign of omega t plus pi pi pi 3.

00:30 So we can clearly see there is a difference of phase here.

00:35 Alright, so that phase difference, pi, is clearly visible.

00:42 This is omega t plus pi by 2, this is omega t plus pi by 3.

00:46 So if you see, if we subtract this phase from this phase, we will get phase difference, which is coming out to be pi by 6 radiance or 30 degrees.

00:59 So we have two waves whose amplitudes are known to us and whose phase difference is also known to us.

01:09 So we can find the resultant definitely.

01:12 So the resultant in this case would be equal to r.

01:18 R is the resultant amplitude.

01:21 Sign of omega t plus pi by 3 because omega t plus pi by 3 is my reference angle.

01:33 And with respect to this reference angle the phase is 30 degree higher you can see the phase of this is 30 degree higher than this right so this is my base function then plus of 10 inverse 10 inverse of the formula is something like this it is b sine p upon a plus b cause 5.

02:10 So this is the formula.

02:12 In this formula, the value of r is important.

02:15 The value of r is also given.

02:17 It is given as a square plus b square plus 2b cause 5 whole root.

02:29 So here a is the amplitude of one phaser.

02:33 B is the amplitude of the second phaser and phi is the phase difference between them...

Please give Ace some feedback