# <font color=#003399 >Wave interference and Energy conservation: facts and misconceptions</font>

* Many of us have wondered about the **energy conservation** happening when the **waves** **interfere** with each other, especially when the interference is destructive, does it **destroy** the energy and **violate** the principle of **energy conservation**?
* In some references, it is suggested that when a destructive interference occurs, it is always accompanied by a constructive interference. In this way, the energy lost at the destructive interference is balanced by the energy gained at constructive interference, avoiding the **violation** of **energy** **conservation**.
* This explaination, however **fails** at particular instances which have been discussed in this article. The article emphasizes on the concept of **work done** by the **source**, which becomes important in power calculations to explain certain situations like **completely constructive interference**. Along with that, the **power equation** for the wave interferences, which holds true in all cases, is introduced.

###### tags: `wave interference` `destructive interference` `conservation of energy` `constructive interference`

## <font color=#CC3300 >Interference: introduction</font> <font color=#515A5A size="4" >*(skip this section if already familiar)*</font>
* When 2 waves reach the same point in space at the same time, the amplitude at that particular point is equal to the vector sum of amplitudes of individual waves. This is also called as the principle of superposition.
* For example, lets consider the simple case of 2 sources emitting waves of **same frequency** on a string, as shown in Figure 1 and 2. The part 1 shows emission from first source, while part 2 shows emission from second source. Part 1+2 shows the resultant wave on the string, which is a vector sum of both the waves.
* If the waves have same amplitude at particular point, they will add up, like in figure 1. We also call this case as constructive interference.
* On the other hand, if the waves have opposite (positive and negative) amplitude at particular point, they will cancel each other, like in figure 2. We also call this case as destructive interference.

**Constructive vs Destructive intereference between 2 waves**

| ![Constructive intereferenc betwn 2 waves](https://i.imgur.com/3r1KYhP.gif "beautiful image" =250x200) | ![](https://i.imgur.com/plCgpaE.gif "beautiful image" =250x200) |
|:--:|:--:|
| *Figure 1: constructive interference*| *Figure 2: destructive interference* |

* All types of waves, for example sound, light, mechanical waves like waves on water follow the principle of superposition.

## <font color=#CC3300 >Energy calculations for wave</font> <font color=#515A5A size="4" >*(skip this section if already familiar)*</font>
* Now let us look at the energy and power calculations for wave interference.
* For a travelling wave of wavelength $\lambda$, amplitude $A$ and angular frequency $\omega$, the total energy associate with one wavelength of the wave is $$E_\lambda = \dfrac{1}{2}{\mu}A^2{\omega^2}\lambda\tag{1}\label{eq1}$$

* Hence if we consider frequency of the wave to be constant, the energy depends on the amplitude as:
$$E_A = KA^2\tag{2}\label{eq2}$$
where $K$ is some constant.

## <font color=#CC3300 >Paradox in energy conservation</font>
* Now let's calculate energy of resultant wave in certain scenarios of interference of 2 waves.
* Suppose 2 sources ($Source_1$ and $Source_2$), each creating a travelling wave of amplitude $A$ and same frequency $f$ are placed on top of each other.
* Individually, energy generated by each source is according to equation $\eqref{eq2}$ is $$E_{Source_1}=E_{Source_2}=KA^2$$. So total energy generated by both sources is $$E_{total}=2KA^2$$
1. When the sources are in phase like in *Figure 1*, at each point in space, their amplitudes will add up, creating a wave of amplitude $2A$. Then the energy associate with this resultant wave will be according to equation $\eqref{eq2}$ $$E_{resultant}=K(2A)^2 = 4KA^2$$
Here, we can easily see the paradox. If the total energy generated by both the sources is $2KA^2$, but the energy of the resultant wave is $4KA^2$, where is the extra energy coming from?
2. When the sources are out of phase like *Figure 2*, at each point in space, their amplitudes will cancel out, creating a wave of amplitude $0$. Then the energy associate with this resultant wave will be according to equation $\eqref{eq2}$ $$E_{resultant}=K(0)^2 = 0$$
Again, where is the energy of $2KA^2$ getting lost? Do these instances violate the energy conservation principle?

## <font color=#CC3300 >Incorrect or Incomplete explaination</font>
* First of all, let us look at the incorrect explaination which states that the energy is getting redistributed, such that the energy which seems lacking at the destructive interference is balanced by increase in energy at constructive interference.

* This explaination works well in certain conditions, for example in **YDSE** or Young's Double Slit Experiment. In YDSE, if the intensity of each light source is say $I_0$, then the total energy of both the sources becomes $2I_0$. When we measure the average intensity at the distribution pattern, it also turns out to be $2I_0$. Hence in the case of YDSE, it can be roughly said that the energy is getting **'redistributed'** from destructive interference to constructive interference.
* One very important observation to be noticed is that in YDSE, **the distance between 2 sources($d$)** is usually in **milimeters** (mm), while the wavelength of source (light) used($\lambda$) is much less, in **nanomaters**. So the ratio **$\dfrac{d}{\lambda}$** is much higher than 1.
* Now let's consider a case where this ratio is less than 1. A case where 2 sources are radiating waves of same frequency and amplitude. In case 1, the distance between these sources is $\lambda/2$, as shown in *Figure 3*, and then it is reduced to $0$ in *Figure 4*. Lets assume the power radiated by each source in the absence of other (isolated source) is $P_0$.

| ![](https://i.imgur.com/6e1QXXY.gif "beautiful image" =400x300) | ![](https://i.imgur.com/KV63xoO.gif "beautiful image" =400x300) |
|:--:|:--:|
| *Figure 3 ([source](https://javalab.org/en/superposition_en/))*| *Figure 4 ([source](https://javalab.org/en/superposition_en/))* |

If we calculate the power dissipated in these 2 cases, it turns out to be equal to **$2P_0$** in case of *Figure 3*, and **$4P_0$** in case of *Figure 4*. This clearly shows that by changing the distance, the net energy dissipated in medium is changing, not conserved.

We can qualitatively see this in *Figure 3* and *Figure 4*. In *Figure 3*, The waves are absent in the sides, and in *Figure 4*, waves are present everywhere. This gives us the clear idea that more energy is being spent in case of *Figure 4*. The question is how?

## <font color=#CC3300 >Explaination of paradox</font>
* In order to give an explanation to this phenomenon, which fits all the cases and doesnt fail at any case, it is necessary to consider the **'work done'** by the source or the **'wave impedance'**. For example, do you think that a source generating a wave of amplitude $A$ in vacuum and a source generating the same wave in the environment when there is a varying electromagnetic field present at the source require same amount of energy?
* This situation can be compared to pushing a rock on a plane surface vs pushing it on a hill. To get the same displacement, more energy needs to be transferred in case of pushing rock on the hill. This is because there is an **'opposing force'** present in the later case.
* In the similar way, when a source is generating a wave of amplitude $A$ when there is already a wave of amplitude $A$ present at the source, it needs to provide more energy to the system. The additional energy it is providing in this case is $KA^2$. So total energy provided by one source becomes $2KA^2$, and when we add the energy from both the sources, the net energy becomes $4KA^2$.

Constructive interference | No interference | Destructive interference
:-------------------------:|:-------------------------:|:-------------------------:
![](https://i.imgur.com/rcuYmKp.png "beautiful image" =150x150)| ![](https://i.imgur.com/1lzh64M.png "beautiful image" =150x150)| ![](https://i.imgur.com/L1hbMDQ.png "beautiful image" =150x150)
![](https://i.imgur.com/SMdkMMZ.png "beautiful image" =150x150) | ![](https://i.imgur.com/vB4d3WT.png "beautiful image" =150x150) | ![](https://i.imgur.com/5dGHIhc.png "beautiful image" =150x150)
higher work done by source compared to isolated source|isolated source (lesser work done compared to source present in constructive interference)| no work done by source

* The case of 2 sources of opposite phase present at same point, can be compared to pushing a rock down the hill. Because of the opposite nature of the field present at the source, the total work done by the source will become $0$, and that is why there is no net radiation present, radiation being the transfer of energy to the medium by the source.
* The concept of work done by source was explained using term **wave impedance** by **Levine** in 1980 [Ref](https://www.tf.uni-kiel.de/matwis/amat/admat_en/articles/interference_paradox_article.pdf) [target=_blank], it's explained in [ref](http://aapt.scitation.org/doi/10.1119/1.4895362) written by Authors Robert Drosd, Leonid Minkin, and Alexander S. Shapovalov

## <font color=#CC3300 >The equation which explains all interference cases</font>
**Vanderkooy** and **Lipshitz** have derived an equation which describes the total power radiated by the 2 sound sources as a function of distance between them. This equation can be also used for other type of waves such as EM wave or transverse waves. The equation is as follows:

$$P=2P_0(1+\dfrac{\sin(kd)}{kd}\cos\theta)\tag{3}\label{eq3}$$

Where,
1. $P$ is total power radiated by both sources
2. $P_0$ is the power radiated by single source in absence of other source
3. $k=\dfrac{2\pi}{\lambda}$, $\lambda$ is wavelength of source
4. $d$ is the distance between 2 sources
5. $\theta$ is the phase difference between 2 sources

It can be better understood graphically, which is shown in Figures 5 and 6
### Graph of equation $\eqref{eq3}$
![](https://i.imgur.com/hgpUhl0.png "beautiful image" =350x300) ![](https://i.imgur.com/QpcYHT8.jpg "beautiful image" =350x300)*Figures 5,6*

### Graph analysis in different cases
Let's understand the graph case by case
* When the 2 sources are in phase, or $cos\theta=1$
1. , the highest total power emitted by both sources is $4P_0$, when the distance between the sources is $0$.This scenario is also shown in [Figure 4](#figure-4).
2. When the distance between 2 sources becomes $\dfrac{\lambda}{2}$ (or $\dfrac{d}{\lambda}=0.5$), the total power radiated drops to $2P_0$. This is also visible in [Figure 3](#figure-3).
3. As the ratio $\dfrac{d}{\lambda}$ becomes >> 1, the value of $P$ tends to converge to $2P_0$. This is what is observed in experiments like YDSE, where the ratio $\dfrac{d}{\lambda}>>1$
* When the 2 sources have phase difference of $\pi$, or $cos\theta=-1$
1. , the lowest total power emitted by both sources is $0$, when the distance between the sources is $0$.
2. When the distance between 2 sources becomes $\dfrac{\lambda}{2}$ (or $\dfrac{d}{\lambda}=0.5$), the total power radiated becomes $2P_0$.
3. As the ratio $\dfrac{d}{\lambda}$ becomes >> 1, the value of $P$ tends to converge to $2P_0$.

Now we can understand why the explaination of **'redistribution of energy'** roughly fits. This is because when $\dfrac{d}{\lambda}>>1$, $P$ tends to converge to $2P_0$. So there is not extra or less work done by the sources, and hence the total energy radiated in medium is same as the energy of individual sources when isolated.

### Questions:
Let's try to answer some common questions that arise in interference with the help of the equation $\eqref{eq3}$.

1. Is the total power provided by 2 interfereing sources always equal to sum of individual sources when isolated ($2P_0$)? :
*Answer*: No. When $\dfrac{d}{\lambda}<=1$, it can vary between $0$ and $4P_0$ depending on phase difference. But then $\dfrac{d}{\lambda}>>1$, it converges to $2P_0$
2. Can there be pure destructive intereference when 2 sources are apart from each other (not placed on top of each other)?
*Answer*: Well this is a topic on its own. But the short answer which can be seen from equation $\eqref{eq3}$ is No, atleast for 3D spherical waves which have been discussed in this article, it is not possible to create purely destructive interference with 2 sources, but the intensity can be significantly reduced when $\dfrac{d}{\lambda}<1$ and $\cos\theta=-1$. Also, total destructive interference can be made possible if the 2 sources are on a string, or in 3D space if they are emitting plane waves.
3. Can there be instances where net radiated power ($P$) is greater than sum of individual sources ($2P_0$) when isolated:
*Answer*: Yes, of course, especially when $\dfrac{d}{\lambda}<1$ and $\cos\theta=1$
4. Can there be instances where net radiated power ($P$) is lesser than sum of individual sources when isolated:
*Answer*: Yes, especially when $\dfrac{d}{\lambda}<1$ and $\cos\theta=-1$
5. What happens to energy when waves interfere?
*Answer*: It depends on the impedance experienced by the sources. For example, if the source experiences high impedance, like in constructive interference, it will do extra work and provide extra energy to the medium. Section [Explaination of paradox](explaination-of-paradox) explains it in detail.
6. Is energy lost on wave interference? or Where does the energy go when waves cancel?
*Answer*: In case of interferences where the net energy is much lesser than the energy provided by the individual sources when isolated, the net work done by the sources is very less, hence energy supplied to the medium becomes less. There is no violation of energy conservation. Section [Explaination of paradox](explaination-of-paradox) explains it in detail.
7. Is energy always conserved when two waves interfere explain?
*Answer*: Although there is no violation to the law of conservation of energy for the whole system, the power received by the medium is not necessarily always the same or conserved. It can vary from $0$ to $4P_0$, where $P_0$ is the power radiated by individual source when isolated (no interference).

## <font color=#CC3300 >To conclude</font>

For energy calculations of wave interferences it is necassary to consider the **'wave impedance'** or the **'work done'** by the source, as without considering these concepts, it is very difficult to explain the gain and loss of energy in certain situations like completely constructive or destructive interference.

It is not perfectly valid to say energy is **'redistributed'** in the wave interference. Although it is **roughly valid** for the case when **$d>>\lambda$** ($d$ is distance between 2 sources and $\lambda$ is wavelength). This assumption **fails** when $d$ is less than $\lambda$.

The **equation $\eqref{eq3}$** given by **Vanderkooy** and **Lipshitz** fits all the cases in a much better way, which explains that the total energy radiated by the 2 interfering sources can vary from **$0$** to **$4P_0$**, depending upon the distance between the sources and phase difference, where $P_0$ is the power radiated by single source when isolated. The excess or lack of power (when $P_{total}>2p_0$ or $P_{total}<2p_0$) can be explained by the **excess** or **lack** of **work done** by the **sources**.

## References
* This article is based on the paper '[Interference and the Law of Energy Conservation](http://aapt.scitation.org/doi/10.1119/1.4895362)' by
Robert Drosd, Leonid Minkin, and Alexander S. Shapovalo, which in turn is based on papers '[False paradoxes of superposition in electric and
acoustic waves](https://www.tf.uni-kiel.de/matwis/amat/admat_en/articles/interference_paradox_article.pdf)' by R.C. Levine and paper '[Power response of loudspeakers with noncoincident drivers – The influence of crossover design](https://www.aes.org/e-lib/browse.cfm?elib=11715)' by J. Vanderkooy and S. P. Lipshitz and more.

---
- [Interference - introduction](#interference:-introduction)(skip this part if already familiar)
- [Energy calculations for waves](#energy-calculations-for-wave)(skip this part if already familiar)
- [Paradox in energy conservation](#paradox-in-energy-conservation)
- [Incorrect or Incomplete explaination](#incorrect-or-incomplete-explaination)
- [Explaination of paradox](##Explaination)
- [Wave impedance](###Wave)
- [Power Calculations](#Power)
- [Analysis in different cases](#analysis)
- [To conclude](#to)
- [References](#references)
---