Let's see what happens to wave equation under Galilean transformations. Electromagnetic wave equation is given by del square phi is equal to 1 upon c square, delta square divided by delta square where phi can be electric field or magnetic field. If one is in the gradient transformations then in S prime I should get del square, del prime square operating on phi equal to 1 by c square divided by 2 phi by divided by p prime square. Now if we write equation 1 component wise, if you are unaware of the operator del square, now we will get the equal to this, that is del prime 2 by del square plus del prime 2 by del plus z square on phi first on my c square divided by 2 phi by del prime t square. Now these are nothing but second order partial derivatives, c is nothing but speed of magnetic wave field that is light in free space. Now chain rule for partial differentiation is del by f by del by x can be written as del by f by del by x prime, del by x prime by del by x plus del by f by del by f by del by del by x and so on for all the variables. Now, c allele transformations are x prime is x minus dt y prime is y z prime is z t prime is p. So, del by x prime by del by x is 1 del by y prime by del by x is 0 to same as del by z prime by del by x and del by p prime by del by x. So, 5 becomes del by f by del by x is same as del by f upon del by x prime which means del by 2 f by del by x where because we want second order derivative when we go to del by del by x into del by f by del by x which can be written as del by del by x prime by del by x prime which is nothing but del by 2 f by del by x prime square. Similarly, for y components del by f by del by y will be equal to del by f by del by x prime del by x prime by del by y plus del by f by del by y prime to del by y prime by del by y and so on. This is 0, this is 1, this is 0, this is 0. So, del by f by del by y same as del by f by del by y prime which means del by 2 f by del by y square will be same as del by 2 f by del by y prime square. So, del by f by del by 2 f by del by z square will come out to be del by 2 f by del by z square you can work it out for self equation 9. Now, del by f by del by t will be del by f by del by x prime del by x prime del by del by t and so on for all variables. Now, if we go and look back to the transformation equations del by x prime by del by t will give me minus v and del by t prime by del by t will give me 1. So, this becomes this is minus v, this is 0, this is 0, this is 1. So, del by f by del by t is nothing but del by del by t dash from here minus v del by del by x prime this term. So, del by 2 f by del by t square will be del by del by t dash minus v del by del by x prime operating on del by del by t prime minus v del by del by del by x prime on f. So, this is the definition del. Now, all these operators you want to substitute in here. So, a substitute equation 7, 8, 9 and 10 into 5. So, this is the definition del by t square plus del by t square plus del by t square plus del by t square plus del by t square plus del by t square. Now, bringing on 5, going to 1 by c square, now del by del by t dash minus v del by del by del by t dash minus v del by del by x dash operating on 5. This is what I am getting, but this is not of the form of del by 2 by del by x prime square plus del by 2 by del by del by del by z prime square operating on 5 equal to 1 by c square, del by 2 by del by t prime square. This is what we wanted, but this is what we are getting. So, these are two different forms. If the equation was invariant, this is what we were expecting, but we are getting this. So, we are not getting the form of del square prime del by t square 5 equal to 1 by c square del by 2 by del by t prime square, which means, very equation is not in the way we are expecting. In the invariant under del invariant transformations.