wavelength
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to be on the same wavelength
In a figurative sense, the way people harmonize with and understand each other.
Wir haben sofort gemerkt, dass wir auf der gleichen Wellenlänge sind.
We noticed right away that we were on the same wavelength.
The word is made up of 'die Welle' (the wave) and 'die Länge' (the length). So the physical meaning is quite literally 'the length of a wave.'
Die Wellenlänge von sichtbarem Licht ist extrem kurz.
The wavelength of visible light is extremely short.
The figurative meaning is used almost exclusively in the fixed expression 'auf der gleichen Wellenlänge sein/liegen' (to be on the same wavelength). The preposition 'auf' with the dative is fixed here.
Mein neuer Kollege und ich sind zum Glück auf der gleichen Wellenlänge.
Luckily, my new colleague and I are on the same wavelength.
You can recognize the physical meaning from a technical or scientific context, such as radio, physics, or light. You can almost always recognize the figurative, personal meaning from the fixed expression 'auf der gleichen/einer Wellenlänge sein' (to be on the same wavelength).
Der Sender hat seine Wellenlänge geändert.
The station changed its wavelength.
understanding
'das Verständnis' and 'die Wellenlänge' can both describe people getting along well and understanding each other. 'Verständnis' means that you can understand another person's feelings, reasons, or situation. 'Auf der gleichen Wellenlänge sein' (to be on the same wavelength) is a figurative expression and means that people think similarly, feel similarly, or harmonize well. They can be confused because both are positive in interpersonal contact. 'Verständnis' fits when the focus is on understanding and consideration. 'Wellenlänge' fits when the relationship or collaboration simply feels like a good match.
Schon im ersten Gespräch waren die neue Kollegin und ich auf der gleichen Wellenlänge.
Already in the first conversation, my new colleague and I were on the same wavelength.
Bei der Arbeit zeigt ihr Verständnis, wenn ein Kollege wegen seines Kindes früher gehen muss.
At work, she shows understanding when a colleague has to leave early because of his child.