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From multiple online sources I read that $$E \propto A^2$$ but when I mentioned this in class, my teacher told me I was wrong and that it was directly proportional to amplitude instead.

As far as I know, every website I stumbled upon concerning this said that is the case. My teacher has a Ph.D and seems pretty experienced, so I don't see why he would make a mistake, are there cases where $E \propto A$?

I also saw this derivation:

$$\int_0^A {F(x)dx} = \int_0^A {kx dx} = \frac{1}{2} kA^2$$

located here, does anyone mind explaining it in a bit more detail? I have a basic understanding of what an integral is but I'm not sure what the poster in the link was saying. I know there is a pretty good explanation here, but it seems way too advanced for me (gave up once I saw partial derivatives, but I see that they're basically the same later on). The first one I linked seems like something I could understand.

From multiple online sources I read that $$E \propto A^2$$ but when I mentioned this in class, my teacher told me I was wrong and that it was directly proportional to amplitude instead.

As far as I know, every website I stumbled upon concerning this said that is the case. My teacher has a Ph.D and seems pretty experienced, so I don't see why he would make a mistake, are there cases where $E \propto A$?

I also saw this derivation:

$$\int_0^A {F(x)dx} = \int_0^A {kx dx} = \frac{1}{2} kA^2$$

located here, does anyone mind explaining it in a bit more detail? I have a basic understanding of what an integral is but I'm not sure what the poster in the link was saying. I know there is a pretty good explanation here, but it seems way too advanced for me (gave up once I saw partial derivatives, but I see that they're basically the same later on). The first one I linked seems like something I could understand.

From multiple online sources I read that $$E \propto A^2$$ however, on a couple of other sources I saw, instead, that $$E \propto A$$

Are there cases where the latter is true? Is the first one definitely correct?

I also saw this derivation:

$$\int_0^A {F(x)dx} = \int_0^A {kx dx} = \frac{1}{2} kA^2$$

located here, does anyone mind explaining it in a bit more detail? I understand the terminology, but not necessarily the full concept behind it. I know there is a pretty good explanation here, but it seems a bit too advanced. The first one I linked seems more like something I could understand.

From multiple online sources I read that $$E \propto A^2$$ but when I mentioned this in class, my teacher told me I was wrong and that it was directly proportional to amplitude instead.

As far as I know, every website I stumbled upon concerning this said that is the case. My teacher has a Ph.D and seems pretty experienced, so I don't see why he would make a mistake, are there cases where $E \propto A$?

I also saw this derivation:

$$\int_0^A {F(x)dx} = \int_0^A {kx dx} = \frac{1}{2} kA^2$$

located here, does anyone mind explaining it in a bit more detail? I have a basic understanding of what an integral is but I'm not sure what the poster in the link was saying. I know there is a pretty good explanation here, but it seems way too advanced for me (gave up once I saw partial derivatives, but I see that they're basically the same later on). The first one I linked seems like something I could understand.

From multiple online sources I read that $$E \propto A^2$$ but when I mentioned this in class, my teacher told me I was wrong and that it was directly proportional to amplitude instead.

As far as I know, every website I stumbled upon concerning this said that is the case. My teacher has a Ph.D and seems pretty experienced, so I don't see why he would make a mistake, are there cases where $E \propto A$?

I also saw this derivation:

$$\int_0^A {F(x)dx} = \int_0^A {kx dx} = \frac{1}{2} kA^2$$

located here, does anyone mind explaining it in a bit more detail? I have a basic understanding of what an integral is but I'm not sure what the poster in the link was saying. I know there is a pretty good explanation here, but it seems way too advanced for me (gave up once I saw partial derivatives, but I see that they're basically the same later on). The first one I linked seems like something I could understand.

From multiple online sources I read that $$E \propto A^2$$ however, on a couple of other sources I saw, instead, that $$E \propto A$$

Are there cases where the latter is true? Is the first one definitely correct?

I also saw this derivation:

$$\int_0^A {F(x)dx} = \int_0^A {kx dx} = \frac{1}{2} kA^2$$

located here, does anyone mind explaining it in a bit more detail? I understand the terminology, but not necessarily the full concept behind it. I know there is a pretty good explanation here, but it seems a bit too advanced. The first one I linked seems more like something I could understand.