Is the One Dimensional Wave Equation Applicable to a String Oscillating in Time?

[praveena]

Hai PF,
I had a doubt in the sector of partial differential equation using one dimensional wave equation. Actually the problems is below mentioned
A string is stretched and fastened at two points x=0 and x=2l apart. motion is strated by displacing the string in the form y=k(2lx-[x][/2]) from which it is released at time t=0.find the displacement of the string at any time 't'.

I had attached a diagram. In that they had mentioned x-axis as string length & y-axis as displacement. Is this possible for a one dimensional equation? For one dimensional equation there may exist one spatial dimension & space(i.e time). But in this problem they had given two co-ordinate axis & they had continue the sum with one dimensonal equation. How this is possible? Can anyone explain?

 

[blue_leaf77]

There are indeed two spatial coordinates in the problem, but they are not independent, therefore there is only one independent quantity we treat this as the x coordinate. One may say then the problem is of one space dimension problem.

 

[praveena]

There are indeed two spatial coordinates in the problem, but they are not independent, therefore there is only one independent quantity we treat this as the x coordinate. One may say then the problem is of one space dimension problem.

Sorry, I can't understand.Is we treat time as x co-ordinate?

 

[blue_leaf77]

In that problem there is only one independent spatial coordinate, which we take as x. Time, of course, is another independent variable. In total there are then two independent variables, x and t. This means the solution of the differential equation should be a function of x and t and you can't express it either as a function of x only or t only.

 

[praveena]

In that problem there is only one independent spatial coordinate, which we take as x. Time, of course, is another independent variable. In total there are then two independent variables, x and t. This means the solution of the differential equation should be a function of x and t and you can't express it either as a function of x only or t only.

"There are indeed two spatial coordinates in the problem, but they are not independent" you had mentioned this line in your 1st reply.but in the second reply you had told
that"In that problem there is only one independent spatial coordinate," What does it mean? You are confusing me?

 

[blue_leaf77]

Ok let me revise my statement. In our problem, the string which is stretched in x direction moves in y direction back and forth (i.e. it's oscillating) in time. This means you can always write the displacement in y direction at a given time as a function of x, so $y=y(x)$. Knowing this it's clear that x and y are not independent quantities, they are dependent on each other. Now we must choose which one of them will be treated as the (independent) variable of the problem, and the common choice is x, having chosen x as a variable of the problem, we may say that we have now established the first independent variable, which is, again, x.

But the system also evolves in time, and time is always independent on spatial coordinates. So there are actually two independent variables, x and t.

 

[praveena]

Ok let me revise my statement. In our problem, the string which is stretched in x direction moves in y direction back and forth (i.e. it's oscillating) in time. This means you can always write the displacement in y direction at a given time as a function of x, so $y=y(x)$. Knowing this it's clear that x and y are not independent quantities, they are dependent on each other. Now we must choose which one of them will be treated as the (independent) variable of the problem, and the common choice is x, having chosen x as a variable of the problem, we may say that we have now established the first independent variable, which is, again, x.

But the system also evolves in time, and time is always independent on spatial coordinates. So there are actually two independent variables, x and t.

Clearly i had understand. This is why we are saying this problem as one dimensional equation. Thank you.