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I have the following problem:

"A blue light, emitting at a wavelength $\lambda = 400 (nm)$ in its rest frame, is mounted to the side of a train traveling at a constant speed of $v = 0.6c$ in a station’s frame. Calculate the wavelength of the light measured by an observer standing on the station platform at the exact moment at which the train passes. You may neglect the distance between the observer and the train."

Now, I am thinking the following:

Classically any Doppler shift type effect arises because different signals need to travel different distance due to relative motion between the emitter and the observer in the direction of the "signal" (wave). In this case, no Doppler shift should arise in the situation when the emitter travels at right angles to the direction of the signal. Which is to say, in classical physics. However, in the case of the relativistic Doppler shift, there are two factors causing the shift: The classical just described and time-dilation. Hence, in our problem with the train, only the second one will be at work.

Therefore: $ \Delta t' = \gamma \Delta t $, where $ \Delta t = 1/f $ etc. This should yield: $$ \lambda_3 = \frac{\lambda_3'}{\gamma} = 320 (nm).$$ Is this alright? I have never solved a similar problem before and I am therefore uncertain. How do I know for sure if it is $ \Delta t' = \gamma \Delta t $ or $ \Delta t = \gamma \Delta t' $?

I have the following problem:

"A blue light, emitting at a wavelength $\lambda = 400 (nm)$ in its rest frame, is mounted to the side of a train traveling at a constant speed of $v = 0.6c$ in a station’s frame. Calculate the wavelength of the light measured by an observer standing on the station platform at the exact moment at which the train passes. You may neglect the distance between the observer and the train."

Now, I am thinking the following:

Classically any Doppler shift type effect arises because different signals need to travel different distance due to relative motion between the emitter and the observer in the direction of the "signal" (wave). In this case, no Doppler shift should arise in the situation when the emitter travels at right angles to the direction of the signal. Which is to say, in classical physics. However, in the case of the relativistic Doppler shift, there are two factors causing the shift: The classical just described and time-dilation. Hence, in our problem with the train, only the second one will be at work.

Therefore: $ \Delta t' = \gamma \Delta t $, where $ \Delta t = 1/f $ etc. This should yield: $$ \lambda_3 = \frac{\lambda_3'}{\gamma} = 320 (nm).$$ Is this alright? I have never solved a similar problem before and I am therefore uncertain. How do I know for sure if it is $ \Delta t' = \gamma \Delta t $ or $ \Delta t = \gamma \Delta t' $?

I have the following problem:

"A blue light, emitting at a wavelength $\lambda = 400$ nm in its rest frame, is mounted to the side of a train travelling at a constant speed of $v = 0.6c$ in a station’s frame. Calculate the wavelength of the light measured by an observer standing on the station platform at the exact moment at which the train passes. You may neglect the distance between the observer and the train"

Now, I am thinking the following:

Classically any Doppler shift type effect arises because different signals need to travel different distance due to relative motion between the emitter and the observer in the direction of the "signal" (wave). In this case, no Doppler shift should arise in the situation when the emitter travels at right angles to the direction of the signal. This is to say, in classical physics. However, in the case of the relativistic Doppler shift, there are two factors causing the shift: The classical just described and time-dilation. Hence, in our problem with the train only the second one will be at work.

Therefore: $ \Delta t' = \gamma \Delta t $, where $ \Delta t = 1/f $ etc. This should yield: $$ \lambda_3 = \frac{\lambda_3'}{\gamma} = 320 nm$$. Is this alright? I have never solved a similar problem before and I am therefore uncertain. How do I know for sure if it is $ \Delta t' = \gamma \Delta t $ or $ \Delta t = \gamma \Delta t' $?

I have the following problem:

"A blue light, emitting at a wavelength $\lambda = 400 (nm)$ in its rest frame, is mounted to the side of a train traveling at a constant speed of $v = 0.6c$ in a station’s frame. Calculate the wavelength of the light measured by an observer standing on the station platform at the exact moment at which the train passes. You may neglect the distance between the observer and the train."

Now, I am thinking the following:

Classically any Doppler shift type effect arises because different signals need to travel different distance due to relative motion between the emitter and the observer in the direction of the "signal" (wave). In this case, no Doppler shift should arise in the situation when the emitter travels at right angles to the direction of the signal. Which is to say, in classical physics. However, in the case of the relativistic Doppler shift, there are two factors causing the shift: The classical just described and time-dilation. Hence, in our problem with the train, only the second one will be at work.

Therefore: $ \Delta t' = \gamma \Delta t $, where $ \Delta t = 1/f $ etc. This should yield: $$ \lambda_3 = \frac{\lambda_3'}{\gamma} = 320 (nm).$$ Is this alright? I have never solved a similar problem before and I am therefore uncertain. How do I know for sure if it is $ \Delta t' = \gamma \Delta t $ or $ \Delta t = \gamma \Delta t' $?