a wave function expressing displacement (y) as a function of its (x) and time (t) is given as y = P log (Qx + Rt)

Which of the following expressions has dimensions different from one another?

(1) yR

(2) PR

(3) R/Q

(4) QR

4.QR

Because log(x) is dimensionless. Here (x)=(Qx+Rt) So [Q][x]=[M^0][L^0][T^0] So [Q]=[L^-1]
Then [R]=[T^-1] & [P]=[L] ( Same as dimension of y= displacement)
Then the only answer is QR which have different dimension from the other three options.

Y=Plog(Qx+e^-Rt)

In log(x) x is dimensionless being a constant
So (Qx+e^-Rt) = 1
Also dimensions of Qx= dimensions of e^-Rt as only similar quantities can be added or subtracted
Hence Qx=1
Q= x^-1=[L^-1]
Similarly, -Rt = 1
R= [T^-1]
Y=p=[L]
Put these values in the options and find the odd one out i.e. QR

and your thinking is?

To determine which of the given expressions have dimensions different from one another, we first need to analyze the dimensions of each of the variables in the wave function.

In the given wave function, y = P log(Qx + Rt), we can see that:
- P is the magnitude of the displacement (y), so it has dimensions of length.
- Qx represents the position (x) multiplied by a constant (Q), so it also has dimensions of length.
- Rt represents the product of time (t) multiplied by a constant (R), so it has dimensions of time.

Now, let's analyze the dimensions of each of the given expressions:

(1) yR:
- y has dimensions of length.
- R is a constant, so it does not have any dimensions.
- Therefore, the dimensions of the expression yR are length.

(2) PR:
- P has dimensions of length.
- R is a constant, so it does not have any dimensions.
- Therefore, the dimensions of the expression PR are length.

(3) R/Q:
- R has dimensions of time.
- Q is a constant, so it does not have any dimensions.
- Therefore, the dimensions of the expression R/Q are time.

(4) QR:
- Q has dimensions of length.
- R is a constant, so it does not have any dimensions.
- Therefore, the dimensions of the expression QR are length.

From the analysis above, we can see that the expressions (1) yR and (2) PR have the same dimensions of length, while the expressions (3) R/Q and (4) QR also have the same dimensions of length.

Therefore, the expressions (1) yR, (2) PR, (3) R/Q, and (4) QR all have dimensions that are the same as one another. None of the expressions have dimensions different from one another.