Two waves having amplitudes of 5 and 8 units and equal frequencies come together at a point in s

Two waves having amplitudes of 5 and 8 units and equal...

Key Concepts

Wave Interference

Wave interference is the phenomenon that occurs when two or more waves overlap in space, leading to the formation of a new wave pattern. This typically involves the superposition of individual wave amplitudes, where the resultant amplitude depends on the phase relationship between the waves. Constructive interference occurs when waves are in phase, resulting in a higher amplitude, while destructive interference happens when the waves are out of phase, reducing the overall amplitude.

Amplitude Addition and Phasor Representation

When combining two waves with a phase difference, the resultant amplitude can be determined using vector (or phasor) addition. Each wave's amplitude is represented as a vector in the complex plane, and their sum is obtained by adding these vectors taking into account the phase difference between them. The magnitude of the resultant vector is given by the square root of the sum of squares of the individual amplitudes plus twice the product of the amplitudes times the cosine of their phase difference.

Intensity as a Function of Amplitude

Intensity in wave phenomena is typically proportional to the square of the wave's amplitude. This means that when calculating intensity, one squares the amplitude of the resultant wave. Comparing the intensity of the combined wave to the sum of the intensities of individual waves involves taking the ratio of the square of the resultant amplitude to the sum of the squares of the individual amplitudes.

Phase Difference

Phase difference refers to the difference in the phase of two waves at a given point in space and time. It is a critical factor in determining the nature of interference between waves. A phase difference influences the cosine term in the amplitude addition formula and therefore the constructive or destructive characteristics of the interference, affecting both the resultant amplitude and the corresponding intensity.

Transcript

00:01 Two waves having amplitudes of five and eight units right? five and eight units so let us say this is a1 and this is a2 right and all right and have equal frequencies come together at a point in space right so now if they meet as a phase difference let us say the five will be equals to five pi by eight radians right or you can also write it in this form degree form that is 180 multiplied by radian is nothing but 180 degrees so this is divided by 8 so what it gives you it gives you the value in degrees it was in radian, right? so it was in radian and the in degrees if you want to find out you just have to multiply by 180 degrees instead of pi right and then you have to divide it by 8 so that would give you the value 112 0 .5 degrees right so this is in degree perfect so now we know intensity i intensity i intensity i is directly proportional to amplitude? square of amplitude right amplitude right this is whole square, right? so what you will do is you are asked to find out the resultant instant intensity i'm sorry resultant intensity relative to the sum of two separate intensities right so now when you're talking about that so i resultant therefore i resultant would be equals to i 1 plus i 2 plus root over i 1 i 2 multiplied by cos 5 isn't it so this is going to be the final formula now we got the relationship between these two that means you can easily find out this value that means this this would be a 1 square can you write it a 1 square plus a 2 square? right plus you have a 1 square e 2 square isn't it so now this would be cos? 5 again, so now it becomes a 1 square means what here a 1 is 5 units and a 2 is 8 units so 5 square plus 8 square plus root over 5 square 8 square multiplied by cos 115 degrees now let us just find out it 25 plus 64 plus 5 multiplied by 8 so it will be taken out that means it becomes 40 cos 115 let us just find out what it gives you it gives you first of all i am finding out the value 115 alright, so i'm sorry it should be 112 point 5.

03:39 I'm sorry it should be 112 point 5 112 point 5 112 point 5 i'm sorry okay, so now it would be cos 112 point 5 right and that would be equals to the value negative of 0 .3826 since you can write here cos 112 point 5 degrees is equals to negative of 0 .3826 8 right so now i'm multiplying it directly by 40 here because we will be getting the same value right so it would be equals to negative of 15 point 307 and we just need to add here 25 then you have to add 64 right so in total it becomes and so i'm so sorry i need to write here to to root i when i do i am correcting myself it should be to root even it oh yes, so now we will solve according to that right so further when you solve it you will be getting 58 .38 5 right so this is in like i resultant so if you talk about intensity in like you can see in units, so it should be 58 .385 so this is going to be the answer this is what we were asked to find out...

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