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What is the meaning of the phase of a wave and phase difference? How do you visualize it?

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Here is a graph of a sine function. It is a function of the angle $\theta$, which goes from $0$ to $2 \pi$, and the value of $\sin(x)$ is bounded by $0$ and $1$.

enter image description here

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

In the following equation

$$u(x, t) = A(x, t)\sin(kx - \omega t + \phi)$$

$\phi$ ("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 overlapping, 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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  • $\begingroup$ What is the physical significance of phase difference? In superposition? $\endgroup$
    – Harsh
    Feb 24 '13 at 17:32
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    $\begingroup$ 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. $\endgroup$
    – 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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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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You may know that the highest point of a wave is known as the "crest" and the lowest one is known as "trough".

Now, take the graph of a sine and cosine functions. You will see the difference that at the origin: i.e. sine is at zero and cosine is at $1$. So, there is a "phase difference" of $\frac{\pi}{2}$ angle.

For better understanding, you may refer to "Understanding Physics Mechanics Part 2 " by DC Pandey.

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  • $\begingroup$ Does it matter if it's negative or positive phase difference? And how do you figure that out? $\endgroup$
    – Harsh
    Feb 23 '13 at 18:49
  • $\begingroup$ 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) $\endgroup$
    – Deiknymi
    Feb 23 '13 at 18:54
  • $\begingroup$ you can take reference for the graph from the ans given below $\endgroup$
    – Deiknymi
    Feb 23 '13 at 19:01
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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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Basically phase is an angle of line joining the origin and the any point on the wave with $x$ axis of our reference frame and the word phase is defined for the single wave function. But the phase difference is defined for two waves. And it tells us information about the resultant shape of waves either it is constructive or destructive or any complex ie irregular. Which can b later Fourier transformed to sines and cosines waves.

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Wave is a periodic motion. There are many different periodic motions. For instance, take a look at an analog clock. Its second hand makes a full circle every 60 seconds. If you take two clocks, then they will usually have a phase difference: their second hands will be circling every 60 seconds, but at any given time they will be pointing to different number of seconds.

On the other hand if you look at the second hand and a minute hand, then there's no point in talking about the phase difference, because they make a full circle at different frequencies: 60 seconds and 60 minutes.

So, in order to talk about phase difference, we should have two waves at the same frequency. When these waves are not perfectly synchronized, we have a phase difference.

One twist. The phase difference is meaningful only within the period of a wave. In case of a second hand of a clock, there's no point in talking about the phase difference of more than 60 seconds. A phase difference of 61 seconds is the same as phase difference of 1 second.

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In addition to the other answers: the phase is a Lorentz scalar. A plane wave is:

$$ \psi(x, t) = A\exp{i\phi(\vec x, t)}$$

where $\phi$ is the phase as a function of position and time:

$$ \psi(x, t) = A\exp{i(\vec k \cdot \vec x - \omega t)} $$.

This can be written as:

$$ \psi(x_{\mu}) = A\exp(ik^{\mu}x_{\mu}) $$

which is manifestly covariant. The phase is:

$$ \phi(x_{\mu}) = k^{\mu}x_{\mu}$$

All inertial observers see the same phase at a given point in space-time, even though they do not agree on frequency or wavelength.

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