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What do you mean by phase of a wave? And phase difference? Waves have always confused me as it's too difficult to visualize them. I am no good at waves mechanics, so if anyone could explain in simpler term? Also if anyone could direct me to complete solution to my problem as to study waves from where? How do you visualize it? How did you study and understand? I have been meaning to ask this question for quite a time but always thought it would be too dumb. Thank you and I am eagerly waiting for an answer and your valuable suggestion

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I take it wikipedia doesn't help? –  Emilio Pisanty Feb 23 '13 at 18:31
    
No. I don't get half of the things on wikipedia. –  Harsh Feb 23 '13 at 18:38
    
yeh wikipedia will not help you till you dont know the basics –  Akash Feb 23 '13 at 18:42
    
Do you know what a sine or a cosine function is? –  anna v Feb 23 '13 at 18:44
    
Yes I know what is sine and cosine function –  Harsh Feb 23 '13 at 18:46
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6 Answers

Here is a graph of a sine function, it is a function of the angle which goes from 0 to two pi and the value of sine(theta) is bounded by 0 and 1.

enter image description here

This function of theta carried on further on the x axis repeats itself every 2pi.

From the graphic one can see that it looks like a wave, and in truth sines ( and cosines) come as solutions of a number of wave equations, where the variable is a function of space and time.

amplitude

here phi ($\phi$) is a "phase" . It is a constant that tells at what value the sine function has when $t=0$ and $x=0$.

If one happens to have two waves ovelapping, then the $\phi_1$-$\phi_2$ of the functions is the phase difference of the two waves. How much they differ at the beginning ($x=0$ and $t=0$), and this phase difference is evidently kept all the way through.

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What is the physical significance of phase difference? In superposition? –  Harsh Feb 24 '13 at 17:32
    
If there are two sine waves and they have a phase difference of pi you can see that by superposing them the two functions will cancel if they have the same amplitude. you can play with this demonstrations.wolfram.com/… to get some intuitive grasp. –  anna v Feb 24 '13 at 17:40
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Let us consider a travelling wave along a very long piece of string. The string will oscillate, and the displacement, $y$, of the string from the flat position (no wave at all) is given by the following equation assuming that the wave does not have a head start

$y(x,t)=A_0\sin(\frac{2\pi}{\lambda}x-\frac{2\pi}{T}t)$

where:

$A_0$ = the maximum departure of the string from the flat position (called: amplitude)

$T$ = the time taken by a particle in the string to complete one oscillation, return to its initial position and repeat the oscillation over and over again.

$\lambda$ = the wavelength of the wave along the string. Imagine this as the distance travelled by the wave in one period, T. Hence one can write the equation $v=\lambda f$, where $f$ is the frequency of the oscillation of a particle in the string. You can thing of this as the number of complete cycles the wave is doing in one second.

The Phase:

The phase of the wave is the quantity inside the brackets of the sin-function, and it is an angle measured either in degrees or radians.

$\phi=(\frac{2\pi}{\lambda}x-\frac{2\pi}{T}t)$

The phase of a wave is not a fixed quantity. Its value depends on what point along the x-axis and at what time you observe the wave. For example, if you consider two points $x_1$ and $x_2$ along the $x$-axis at some common instant in time $t_c$, these two points will have their own phase $\phi_1$ and $\phi_2$ given as

$\phi_1=( \frac{2\pi}{\lambda}x_1-\frac{2\pi}{T}t_c)$

$\phi_2=(\frac{2\pi}{\lambda}x_2-\frac{2\pi}{T}t_c)$

The phase difference the wave has at these two points is

$\phi_2-\phi_1= \frac{2\pi}{\lambda}x_2 -\frac{2\pi}{\lambda}x_1 $

$\phi_2-\phi_1=\frac{2\pi}{\lambda}(x_2-x_1)$

The important result here is that the two waves can be:

(1) In phase if $x_2-x_1=n\lambda$, i.e the wave is doing exactly the same thing at such points along the x-axis.

(2) Out of phase if $x_2-x_1=(n+\frac{1}{2})\lambda$, i.e one point in the string, $x_1$ say, is moving upwards while $x_2$ is moving downwards but symmetrically.

This analysis holds for two coherent waves coming from two coherent sources, travelling different distances and combine at some point that is distance $x_1$ from one source and distance $x_2$ from the other source. So you will get constructive interference in case (1), and destructive interference in case (2). This is why you are able to observe the interference pattern.

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You would probably know the shape of waves so the higher point is known as crest and the lower one is known as trough. Ok now take the graph of a sign function and a cos function you will see the difference that at the origin sin is at zero and cos is at 1 so there is a phase difference of pie/2 angle because when you will draw the cos graph towards negative x axis you will see it touches the x axis at -(pie)/2. For better understanding you may refer to arihant mechanics part 2 by dc pandey . Hope it helped you. And never fell yourself dumb in asking questions as you dont know how hard is the topic it also took me about a year to figure out

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Does it matter if it's negative or positive phase difference? And how do you figure that out? –  Harsh Feb 23 '13 at 18:49
    
Its just depend on frame of reference for example if take +X as positive and -X axis as negative then if wave has a phase difference of +0 then it mean's it travelling 0 angel forwrdthan the other wave and vice versa for ex let wave 1 be Asin(wt) and 2 be Asin(wt + 0) then two is moving 0 angel's forward than 1 and if 2 = Asin(wt - 0) then it is moving 0 angel's behind to 1 (i have use 0 for simbolising theta) –  Akash Feb 23 '13 at 18:54
    
you can take reference for the graph from the ans given below –  Akash Feb 23 '13 at 19:01
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I think the relevant question here is "What is a wave?". We generally define anything that solves the wave equation or generalisations thereof to be a wave; I realise however that may not be quite illuminating.

Fortunately the solutions themselves are easy enough to describe: they are of the form* $$f(x,t)=A\sin\left(\frac{2 \pi}{\lambda} x - \frac{2 \pi}{T} t\right)$$ so that $\lambda$ is the wavelength, $T$ is the period of the wave, and $f(x,t)$ is equal to the amplitude of the wave at the point $x$ at time $t$, while $A$ is a constant. The argument of this function, $$\varphi(x,t)=\frac{2 \pi}{\lambda} x - \frac{2 \pi}{T} t$$ is the phase. With this definition we can write $$f(x,t)=f(\varphi(x,t))$$ so as to consider $f$ as a function of its phase alone. So two waves $f_1$ and $f_2$ have a phase difference $\Delta \varphi$ if $$f_1(x,t)=f_1(\varphi(x,t) + \Delta \varphi) \\ f_2(x,t)=f_2(\varphi(x,t))$$ So when you add sinusoidal waves like these what you get is dependent on $\Delta \varphi$. E.g if you consider the sum $f(\varphi(x,t)+\Delta \varphi) + f(\varphi(x,t))$ with $f$ as above for different values of $\Delta \varphi$ you get

for $\Delta \varphi=0$, $f(\varphi(x,t)+0) + f(\varphi(x,t))=2 f(\varphi(x,t)) $

for $\Delta \varphi=\pi$, $f(\varphi(x,t)+\pi) + f(\varphi(x,t))=-f(\varphi(x,t)) +f(\varphi(x,t))=0$ since $\sin(\phi +\pi)=-\sin(\phi)$ for all $\phi$.

Note that this generalises to other waveforms; for instance you can try $f(\varphi) = e^{-\varphi^2}$ where $\varphi$ is as above and I got tired of writing the $(x,t)$ dependence explicitly :-), and see what you get.

*Of course these are not the only solutions, but any of them can be obtained as a superposition of sinusoids like this. Phase is mostly useful when you are talking about sinusoids or things that resemble them closely enough however.

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What is the meaning of phase difference?

It's an offset, in time or space, of one wave with respect to another

enter image description here

If you make an arbitrary choice and say your wave "starts" when it's height is 0, then if you start a second wave a short time later it will be out of phase with the first wave. If you start the second wave at a later time that is an exact multiple of the time the first wave takes to repeat, the second wave will be in phase.

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Another way to gain insight is to defer the motion of waves and focus on the complex plane and the notion of phasor obtained via Euler's formula, $e^{i \theta} = cos(\theta)+ i sin(\theta) $ (refer to the figure in Wikip).

The set {$e^{i \theta} | \theta \in [0,2 \pi]$} is the unit circle, and the phase difference between any two points on it, eg, $e^{i \phi}$ and $e^{i \psi}$ is well defined and is is simply $\phi - \psi$ (the Wikip. article explains the complex conjugate multiplication involved).

To relate back to waves, replace the constant $e^{i \theta}$ with a function of time $e^{i \omega t} $ where $\omega$ is the angular velocity.

Finally note that two such phasors $e^{i \omega_1 t}$ and $e^{i \omega_2 t}$ do not have a (constant) phase difference if their angular velocities differ, ie if, $\omega_1 \neq \omega_2$, (though rational ratios of angular velocities result in stable winding numbers or entrainment, which is a more general form of phase relation).

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