*Before you read this, I suggest that you read* posts 17.4 and 18.10.

In post 18.10 we saw that a one-dimensional wave could be represented by a wave function, *ψ*, that is a function (post 19.10) of position, *x*, and time, *t*. At a fixed point in space (*x* is constant), the wave function is given by

*ψ* = *A*sin(2π*t*/*T*) = *A*sin(2π*ft*) = *A*sin(*ω**t*). (1)

At a fixed time

*ψ* = *A*sin(2π*x*/*λ*) = *A*sin(2π*kx*) = *A*sin(*κ**x*). (2)

In equations 1 and 2, *A* represents the amplitude of the wave whose time period (post 18.10) is *T* and whose wavelength (post 18.10) is *λ*.

We can find the rate of change of *ψ* by differentiating (post 17.4) equation 1 with respect to time. We can perform an analogous operation in space by differentiating equation 2 with respect to *x*. But, when we differentiate equation 1, we are assuming that *x* is constant. And, when we differentiate equation 2, we are assuming that *t* is constant.

When we differentiate something that is a function of more than one variable, by assuming that all but one of the variables is a constant, the operation is called *partial differentiation*. To show that the result is a partial derivative we replace the usual d that appears in differentiation by *∂* (we call this symbol “partial d”).

Using the result for differentiating sines from appendix 1.1 of post 17.13 gives

*∂ψ*/*∂t *= *ωA*cos(*ωt*) and *∂ψ*/*∂x *= *κ**A*cos(*κ**x*).

That was easy because we started with an expression for *ψ* when *x* is constant and an expression for *ψ* when *t* is constant (equation 2).

Usually we want to perform partial differentiation when our starting equation contains three or more variables. For example, let’s think about the volume, *V*, of an ideal gas (post 18.25) containing *n* moles (post 17.48) at a pressure *p* and temperature (on the Kelvin scale, post 16.34), *T*. Be careful – I’m now using *T* to represent something completely different in the second part of this post. Normally, I would avoid doing this but *T* is so commonly used to represent the time period of a wave and temperature on the Kelvin scale, that I’ve used it to represent both. According to the ideal gas equation (post 18.25)

*V* = *nRT*/*p* = *nRTp*^{-1} (1)

We can calculate the effect on *V* of changing *T *by assuming that *p* is constant and calculating

*∂V/**∂T *= *nRp*^{-1} = *nR*/*p. *(2)

We can calculate the second partial derivative (post 17.4) of *V* with respect to *T* as

*∂*^{2}*V/**∂T*^{2} = 0* *(3)

since, in this operation, we are assuming that *p* is constant. If you’re not sure about how equations 2 and 3 were calculated, see post 17.4.

Similarly, we can calculate the effect on *V* of changing *p *by assuming that *T* is constant and calculating

*∂V/**∂p *= –*nRTp*^{-2}* = *–*nRT*/*p*^{2}*. *(4)

And we can calculate the second partial derivative (post 17.4) of *V* with respect to *p* as

*∂*^{2}*V/**∂p*^{2} = *nRTp*^{-3}* = nRT*/*p*^{3}*. *(5)

If you’re not sure about how equations 4 and 5 were calculated, see post 17.4.

We can also calculate:

Notice that equations 6 and 7 give the same results.

The ideas developed here will be useful when we use differential equations (post 19.10) to describe things that depend on time and space – like diffusion (post 18.26) and wave motion (post 18.10).

__Related posts__

19.10 Differential equations

18.17 Euler’s relation, oscillations and waves

18.16 The square root of minus 1 and complex numbers

18.15 More about exponential growth: the number *e*

18.14 Wave shapes – Fourier series

18.3 Logarithms

17.37 More about torque – cross products of vectors

17.36 More about work – line integrals

17.19 Calculating distances from speeds – integration

17.4 Displacement, velocity and acceleration

17.3 Three-dimensional vectors